Answer:
The family paid $55.12 in sales tax. TOTAL= $148.82 + $55.12
Step-by-step explanation:
IMPORTANT FORMULA: To convert a percent to a number, you need to divide by 100.
Therefore, 8% = 8/100 = 4/50 = 2/25
The family paid a total of $689 without sales tax or tips.
8% of $689 would equal the amount that the family paid for sales tax.
We already found what 8% is equal to and keep in mind that the word "of" means multiplication.
2/25 * $689 = sales tax
$55.12 = sales tax
PART B) Now we have to find the total amount the family paid.
We have to add $689 with sales tax and tips.
We have to find tips.
Using the formula To convert a percent to a number, you need to divide by 100.
20% = 20/100 = 2/10 = 1/5
1/5 * ($689 + $55.12) = tips
tips = $148.82
TOTAL= $148.82 + $55.12
What is the mean of the data?
Answer:
The mean (average) of a data set is found by adding all numbers in the data set and then dividing by the number of values in the set. The median is the middle value when a data set is ordered from least to greatest. The mode is the number that occurs most often in a data set.
Find the area of this circle. Use 3 for T.
Α = πη2
5 in
[?] in?
Hope this help!!!
Have a nice day!!!
The number of adults who attend a music festival, measured in hundreds of people, is represented by the function a(d)=−0.3d2+3d+10, where d is the number of days since the festival opened.
The number of teenagers who attend the same music festival, measured in hundreds of people, is represented by the function t(d)=−0.2d2+4d+12, where d is the number of days since the festival opened.
What function, f(d) , can be used to determine how many more teenagers than adults attend the festival on any day?
f(d)=−0.1d2+d+22
f(d)=0.1d2+d+2
f(d)=−0.1d2+7d+2
f(d)=0.1d2+7d+2
Answer:
f(d)=0.1d^2+d+2
Step-by-step explanation:
t(d)=−0.2d2+4d+12
a(d)=−0.3d2+3d+10
how many more teenagers than adults attend the festival on any day?
==>
f(d) = t(d) - a(d)
=0.1d^2+d+2
You deposit $10,000 in an account earning 4% interest compounded monthly. a. How much will you have in the account in 25 years? b. How much interest will you earn?
Answer:
In 25 years I will have $ 27,137.65. Therefore, I will earn $17,137.65 interest.
Step-by-step explanation:
Given that I deposit $ 10,000 in an account earning 4% interest compounded monthly, to determine how much will you have in the account in 25 years and how much interest will I earn, the following calculation must be performed:
10,000 x (1 + 0.04 / 12) ^ 25x12 = X
10,000 x (1 + 0.00333) ^ 300 = X
10,000 x 2.7137 = X
27,137.65 = X
Therefore, in 25 years I will have $ 27,137.65. Therefore, I will earn $17,137.65 interest.
please I need step by step answer. facyorise a2+8a+15
Answer:
(+3)(+5)
Step-by-step explanation:
Let's factor a2+8a+15
a2+8a+15
The middle number is 8 and the last number is 15.
Factoring means we want something like
(a+_)(a+_)
Which numbers go in the blanks?
We need two numbers that...
Add together to get 8
Multiply together to get 15
Can you think of the two numbers?
Try 3 and 5:
3+5 = 8
3*5 = 15
Fill in the blanks in
(a+_)(a+_)
with 3 and 5 to get...
(a+3)(a+5)
PLEASE WILL MARK IF YOU HELP!!
Answer:
22°
63°
m<H=22°
m<G=63°
Problem 1
Answer: 79--------------------------
Work Shown:
For any triangle, the three angles always add to 180
For any isosceles triangle, the base angles are congruent. The base angles are opposite the congruent sides. We see that angle O = angle H.
O+H+T = 180
H+H+T = 180
2H+T = 180
2H+22 = 180
2H = 180 - 22
2H = 158
H = 158/2
H = 79
=======================================================
Problem 2
Answer: 54--------------------------
Work Shown:
We'll use the same ideas as problem 1.
In this case, angle O = angle D = 63 since they are the base angles opposite the congruent sides.
D+G+O = 180
63+G+63 = 180
G+126 = 180
G = 180-126
G = 54
3. Find the value of X (In the picture) (giving points to best answer/brainlest)
Answer:
101 =x
Step-by-step explanation:
The measure of the exterior angle is equal to the sum of the opposite interior angles
143 = 42+x
Subtract 42 from each side
143-42 = 42+x-42
101 =x
Answer:
x = 101 degrees
Step-by-step explanation:
The sum of the external angle and its adjacent is 180 degrees
143 + y = 180
y = 37 degrees
The sum of the inner angles of a triangle is 180 degrees
37 + 42 + x = 180
79 + x = 180
x = 101 degrees
PLEASE HELP!!!
WILL MARK BRAINLIEST!!!
If the diameter of the circle shown below is 6ft and 0 is a right angle, what is the length of segment AB to the nearest foot?
Multiple choice!
Thank you!
Answer:
how old are you gghhjjzetstu9u
Answer:
4 ft
Step-by-step explanation:
let's find radius first
radius=diameter/2
=6/2
=3 ft
radii=3 ft
Now by using pythagoras theorem
a^2 + b^2 = c^2
3^2 + 3^2 =AB^2
9+9=AB^2
18=AB^2
[tex]\sqrt{18}[/tex] AB
4.24 =AB
4 ft =AB (after converting to nearest foot)
Lori downloaded all the pictures she took at Rita’s wedding into a single computer folder. She took 86 of the 134 pictures with her camera and the remainder of them with her cell phone. Of the pictures Lori took with her cell phone, one out of every five was blurry.
Answer:
87
Step-by-step explanation:
A
2x+5
x² + 5x + 6
x² + 5x+6
B
2x+5
Answer:
what is the question?
Step-by-step explanation:
answer the question
Which of these statements is correct? The system of linear equations 6 x minus 5 y = 8 and 12 x minus 10 y = 16 has no solution. The system of linear equations 7 x + 2 y = 6 and 14 x + 4 y = 16 has an infinite number of solutions. The system of linear equations 8 x minus 3 y = 10 and 16 x minus 6 y = 22 has no solution. The system of linear equations 9 x + 6 y = 14 and 18 x + 12 y = 26 has an infinite number of solutions
Answer:
The only true statement is:
"The system of linear equations 8x - 3y = 10 and 16x - 6y = 22 has no solution."
Step-by-step explanation:
First, some definitions.
A system of linear equations has infinite solutions if both equations define the same line, has no solutions if we have two parallel lines, has one solution in all the other cases.
Where two lines are parallel if we can write them as:
a*x + b*y = c
a*x + b*y = d
where c and d are different numbers.
Now we can analyze the given statements:
a)
6x - 5y = 8
12x - 10y = 16
has no solution?
If we divide both sides of the second equation by 2, we get:
(12x - 10y)/2 = 16/2
6x - 5y = 8
We get the first equation, then both equations define the same line, thus the system has infinite solutions, then the statement is false.
b)
7x + 2y = 6
14x + 4y = 16
has infinite solutions?
Let's divide the second equation by 2, then we get:
(14x + 4y)/2 = 16/2
7x + 2y = 8
If we rewrite our system of equations, we get:
7x + 2y = 6
7x + 2y = 8
These are parallel lines, thus, this system has no solutions.
So the statement is false.
c)
8x - 3y = 10
16x - 6y = 22
has no solution?
Again, let's divide the second equation by 2 to get:
(16x - 6y)/2 = 22/2
8x - 3y = 11
If we rewrite our system:
8x - 3y = 10
8x - 3y = 11
These are parallel lines, thus the system has no solutions, so this statement is correct.
d)
9x + 6y = 14
18x + 12y = 26
Has infinite solutions?
Dividing the second equation by 2 we get:
(18x + 12y)/2 = 26/2
9x + 6y = 13
So the equations are different (are parallel lines again) so this system has not infinite solutions.
Then the statement is false.
Answer:
The answer to your question is the third choice.
Step-by-step explanation:
a) 6x - 5y = 8
12x - 10y = 16
We observe that these lines are the same so they have infinite solutions.
b)
7x + 2y = 6
14x + 4y = 16
These lines are parallel because they have the same slope, so they do not cross, there is no solution.
c)
8x - 3y = 10
16x - 6y = 22
These lines are parallel because they have the same slope, so they do not cross, there is no solution.
d)
9x + 6y = 14
18x + 12y = 26
These lines are parallel because they have the same slope, so they do not cross, they do not have an infinite number of solutions.
SOMEONE HELP ME PLEASE
find the real fifth root of -32
Answer: -2
This is because (-2)^5 = -32. Applying the fifth root to both sides lets us say [tex]-2 = \sqrt[5]{-32}[/tex]
There are four other roots but they are complex. Effectively, we are solving the equation [tex]x^5 + 32 = 0[/tex]
What is the zero of the function represented by this graph?
Use implicit differentiation to find an equation of the tangent line to the curve at the given point. y2(y2 − 4) = x2(x2 − 5) (0, −2) (devil's curve)
Answer:
Step-by-step explanation:
Given that:
[tex]y^2 (y^2-4) = x^2(x^2 -5)[/tex]
at point (0, -2)
[tex]\implies y^4 -4y^2 = x^4 -5x^2[/tex]
Taking the differential from the equation above with respect to x;
[tex]4y^3 \dfrac{dy}{dx}-8y \dfrac{dy}{dx}= 4x^3 -10x[/tex]
Collect like terms
[tex](4y^3 -8y)\dfrac{dy}{dx}= 4x^3 -10x[/tex]
[tex]\dfrac{dy}{dx}= \dfrac{4x^3 -10x}{4y^3-8y}[/tex]
Hence, the slope of the tangent line m can be said to be:
[tex]\dfrac{dy}{dx}= \dfrac{4x^3 -10x}{4y^3-8y}[/tex]
At point (0,-2)
[tex]\dfrac{dy}{dx}= \dfrac{4(0)^3 -10(0)}{4(-2)^3-8-(2)}[/tex]
[tex]\dfrac{dy}{dx}= \dfrac{0 -0}{4(-8)+16}[/tex]
[tex]\dfrac{dy}{dx}= 0[/tex]
m = 0
So, we now have the equation of the tangent line with slope m = 0 moving through the point (x, y) = (0, -2) to be:
(y - y₁ = m(x - x₁))
y + 2 = 0(x - 0)
y + 2 = 0
y = -2
Jai bought a helmet and a pair of skates.
The helmet cost £45.
He sold both items for £224.
Jai made a 120% profit on the cost of the helmet and a 40% profit on the total cost.
What was the percentage profit on the skates?
Give your answer to 1 decimal place.
Answer:
Profit % on skates = 8.7 %
Step-by-step explanation:
Step 1 : Find cost price of skates
Cost price of helmet = £45
Let cost price of skate be = x
Selling price = £224
Cost price = (x + 45)
Total profit % = 40%
[tex]Profit \% = \frac{Selling \ price - cost \ price }{Cost \ price} \times 100[/tex]
[tex]\frac{40}{100} = \frac{224 - (x + 45)}{(x + 45)}\\\\40(x+ 45) = 100(224 - (x +45))\\\\40(x + 45) = 22400 - 100(x + 45)\\\\40(x +45) + 100(x+ 45) = 22400\\\\140(x + 25) = 22400\\\\x + 45 = \frac{22400}{140}\\\\x = 160 - 45 = \£ \ 115[/tex]
Total cost price = 45 + 115 = £160
Step 2 : Selling price of Helmet
Cost price of Helmet = £45
Let selling price of helmet be = y
Profit % of helmet = 120 %
[tex]Profit \% = \frac{selling \ price - cost \ price}{cost \ price}[/tex]
[tex]\frac{120}{100} = \frac{y -45}{45}\\\\\frac{120 \times 45}{100} = y -45\\\\54 = y - 45\\\\99 = y[/tex]
Step 3 : Selling price of skates
Total selling = selling price of helmet + selling price of skates
224 = 99 + selling price of skates
224 - 99 = selling price of skates
125 = selling price of skates
Step 4 : Profit percentage on skates
Cost price of skate = £ 115
Selling price of skate = £ 125
[tex]Profit \% \ on \ skates = \frac{selling\ price- cost \ price }{cost \ price} \times 100[/tex]
[tex]= \frac{125-115}{115} \times 100\\\\=\frac{10}{115} \times 100\\\\= 8.7 \%[/tex]
Use the slope formula to find the slope of the line through the points (2,10) and (10,−8).
The slope formula is the changes of two y-values over/to the changes of two x-values.
[tex] \large \boxed{m = \frac{y_2 - y_1}{x_2 - x_1} }[/tex]
Substitute two given points in the formula to find the slope. The m-term represents the slope from y = mx+b.
[tex]\large{m = \frac{10 - ( - 8)}{2 - 10} } \\ \large{m = \frac{10 + 8}{ - 8} } \\ \large{ m = \frac{18} { - 8} \longrightarrow \frac{9}{ - 4} } \\ \large \boxed{m = - \frac{9}{4} }[/tex]
Answer
The slope is -9/4.Hope this helps and let me know if you have any doubts!
Answer:
m=-9/4
Step-by-step explanation:
Hi there!
The formula for the slope (m) calculated from two points is given as (y2-y1)/(x2-x1), where (x1,y1) and (x2,y2) are points
we are given the two points (2,10) and (10,-8)
to avoid any confusion, let's label the values of the points
x1=2
y1=10
x2=10
y2=-8
now substitute into the formula:
m=(-8-10)/(10-2)
subtract
m=(-18)/(8)
simplify (reduce to lowest terms)
m=-9/4
Hope this helps!
m.ng giúp mình về phần vector trong ma trận nha
Answer:
maybe if u translate it in English
Step-by-step explanation:
it wouldv been helpful if u mind?
A golfer hits a golf ball.
The function
d(t) = –2t2 + 7t + 4
most closely represents the height(h) of the golf ball in feet after t seconds. How
long is the golf ball in the air?
Answer:
The golf ball was in the air for 4 seconds.
Step-by-step explanation:
Solving a quadratic equation:
Given a second order polynomial expressed by the following equation:
[tex]ax^{2} + bx + c, a\neq0[/tex].
This polynomial has roots [tex]x_{1}, x_{2}[/tex] such that [tex]ax^{2} + bx + c = a(x - x_{1})*(x - x_{2})[/tex], given by the following formulas:
[tex]x_{1} = \frac{-b + \sqrt{\Delta}}{2*a}[/tex]
[tex]x_{2} = \frac{-b - \sqrt{\Delta}}{2*a}[/tex]
[tex]\Delta = b^{2} - 4ac[/tex]
In this question:
We have to find the amount of time it takes for the ball to hit the ground. We have that:
[tex]d(t) = -2t^2 + 7t + 4[/tex]
Which is a quadratic equation with [tex]a = -2, b = 7, c = 4[/tex].
How long is the golf ball in the air?
We have to find t for which [tex]d(t) = 0[/tex]
So
[tex]-2t^2 + 7t + 4 = 0[/tex]
[tex]\Delta = b^{2} - 4ac = (7)^2 - 4(-2)(4) = 81[/tex]
[tex]t_{1} = \frac{-7 + \sqrt{81}}{2*(-2)} = -0.5[/tex]
[tex]t_{2} = \frac{-7 - \sqrt{81}}{2*(-2)} = 4[/tex]
Time is a positive measure, so t = 4.
The golf ball was in the air for 4 seconds.
A triangle has three vertices at the points A = (0, -2), B = (3, 4), and C = (-1.5, 2.5). Prove that this triangle is a right-angle triangle, using algebra. (Note: a right angle triangle has one 90 degree angle).
Answer:eraf
Step-by-step explanation:
PLEASE HELP ME I HAVE TO PASS THIS TEST
30 POINTS
Answer:
Hi, there the answer is
These are the equations with exactly one solution
-5x + 12 = –12x – 12
-5x + 12 = 5x + 12
-5x + 12 = 5x – 5
Hope This Helps :)
Step-by-step explanation:
First degree equations
A first degree equation has the form
ax + b = 0
There are some special cases where the equation can have one, infinitely many or no solution
If , the equation has exactly one solution
If a=0 and b=0 the equation has infinitely many solutions, because it doesn't matter the value of x, it will always be true that 0=0
If a=0 and the equation has no solution, because it will be equivalent to b=0 and we are saying it's not true. No matter what x is, it's a false statement.
We have been given some equations, we only need to put them in standard form
-5x + 12 = –12x – 12
Rearranging
7x + 24 = 0
It has exactly one solution because a is not zero
.......................
-5x + 12 = 5x + 12
Rearranging
-10x + 0 = 0
It has exactly one solution because a is not zero
.......................
-5x + 12 = 5x – 5
Rearranging
-10x + 17 = 0
It has exactly one solution because a is not zero
.......................
-5x + 12 = -5x – 12
Rearranging
0x + 24 = 0
It has no solution, no matter what the value of x is, it's impossible that 24=0
Answer: These are the equations with exactly one solution
-5x + 12 = –12x – 12
-5x + 12 = 5x + 12
-5x + 12 = 5x – 5
To start a shop Rajeev and sneha invested $2,75,280 and $5,35,870 .if they needed $ 8,85,356 .How much money should they need more ?
Answer:
$74,206
Step-by-step explanation:
Total money : $2,75,280 + $5,35,870 = $8,11,150
Amount needed : $8,85,356 - $8,11,150 = $74,206
the ratio of the length of a rectangular floor to its width is 3:2 if the length of the floor is 12 meters what is the perimeter of the floor in meters
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Answer:
40 meters
Step-by-step explanation:
The ratio of width to length is 2/3, so the width of the floor is ...
width = (2/3)(12 m) = 8 m
The perimeter is found from ...
P = 2(L +W)
P = 2(12 m +8 m) = 40 m
The perimeter of the floor is 40 m.
_____
Additional comment
As above, the perimeter is twice the sum of length and width. In terms of ratio units, that is p = 2(3 +2) = 10. The length is 3 ratio units, so the perimeter is 10/3 times the length. (10/3)(12 m) = 40 m.
Longhorn Pizza has the following number of topping options available: four vegetables, two meats, and two cheeses. A pizza is ordered with exactly four toppings. What is the probability that the pizza is ordered with exactly two vegetables, one meat, and one cheese
Answer:
The probability is [tex]\frac{24}{70}[/tex].
Step-by-step explanation:
topping options available: four vegetables, two meats, and two cheeses
Number of topping on one pizza = 4
Getting two vegetables = (4 C 2)
Getting one meat = (2 C 1)
Getting one cheese = (2 C 1)
Choosing 4 toppings out of 8 = (8 C 4)
probability that the pizza is ordered with exactly two vegetables, one meat, and one cheese
[tex]\frac{(4C2)\times (2C1)\times (2C1)}{(8C4)}\\\\\frac{6\times 2\times 2}{70}\\\\\frac{24}{70}[/tex]
Which statements are true of the function f(x) = 3(2.5)x? Check all that apply.
The function is exponential.
The initial value of the function is 2.5.
The function increases by a factor of 2.5 for each unit increase in x.
The domain of the function is all real numbers.
The range of the function is all real numbers greater than 3.
Answer:
A) The function is exponential.
C) The function increases by a factor of 2.5 for each unit increase in x.
D) The domain of the function is all real numbers.
Step-by-step explanation:
Got it right on Edge :)
Answer:
a,c,d are correct
Step-by-step explanation:
Patel squeezed oranges so that his family could have fresh-squeezed juice for breakfast. He squeezed StartFraction 4 over 17 EndFraction cups from the first orange, StartFraction 3 over 10 EndFraction cups from the second orange, StartFraction 9 over 20 EndFraction cups from the third orange, StartFraction 3 over 11 EndFraction cups from the fourth orange, and StartFraction 7 over 15 EndFraction cups from the fifth orange. Patel estimates that he needs 3 cups of orange juice for his family. About how much more orange juice does he need to reach his estimate?
Answer:
A. 1/2 cups
Step-by-step explanation:
13/15 is close to 1
1/5 is a small amount
9/20 is just over 1/2
5/11 is just under 1/2
7/15 is just under 1/2
Estimate: 1 + 1/2 + 1/2 + 1/2 + a little = 2 1/2
He needs 3 cups, so he needs another 1/2 cup.
Answer: A. 1/2 cups
Answer:
it a 1/2
Step-by-step explanation:
Find the surface area of each solid figure
Answer:
First find the SA of the triangular figure
4 x 3 = 12 cm^2 (the triangles on the sides)
2 x 3 = 6 cm^2 (the back square)
2 x 5 = 10 cm^2 (the slanted square)
*I'm not sure if this question includes the bottom of the triangle but here it is anyways
4 x 2 = 8 cm^2
Including the bottom the SA of the triangular figure is:
12 + 6 + 10 + 8 = 36 cm^2
Find the SA of the rectangular shape
4 x 2 = 8 cm^2 (the bottom square)
2 x 6 = 12 x 2 = 24 cm^2 (the sides)
4 x 6 = 24 x 2 = 48 cm^2 (the front and back)
Add them up
8 + 24 + 48 = 80 cm^2
If you wanted to find the SA of the whole figure it would be:
12 + 6 + 10 + 8 + 24 + 48 = 108 cm^2
Hope this helps!
Please help show the steps
Please put 15 years old
Answer:
P = $98.77
Step-by-step explanation:
FV = p (1+i)^n -1
i
pv = 700,000
i = .075/12 = .00625
n = (66 - 15)* 12 = 612
700,000 = P (( 1 + .00625)^ 612 -1 /.00625
4375 = P (1.00625)^612 -1)
P = $98.77
Answer:
page 1:
51 years
$98.78
639546.64 (i think)
Page 2:
213 months
17.8 years
321 months
26.8 years
1128.9 months
88.8 years
I would probably choose the second plan because it's rather unlikely that i live past 90
Step-by-step explanation:
page 1
Let's assume the payments are at the end of the month
66-15= 51 years
effective rate: .075/12=.00625
[tex]700000=x\frac{(1+.00625)^{51*12}-1}{.00625}\\x=98.77973387[/tex]
which i guess we can round to 98.78
700000-98.78*(51*12)= 639546.64
This number is really really high and so maybe you want to double check it
page 2
effective rate: .051/12=.00425
[tex]700000=5000\frac{1-(1+.00425)^{-n}}{.00425}\\.405=(1+.00425)^{-n}\\log_{1.00425}.405=-n\\n=213[/tex]
213 months
213/12= 17.8 years
[tex]700000=4000\frac{1-(1+.00425)^{-n}}{.00425}\\.25625=(1.00425)^{-n}\\log_{1.00425}.25625\\n=321[/tex]
321 months
321/12=26.8 years
[tex]700000=3000\frac{1-(1+.00425)^{-n}}{.00425}\\.008333333=(1.0045)^{-n}\\log_{1.0045}.00833333=-n\\n=1128.9[/tex]
1128.9 months
1128.9/12= 94.1 years
1066 months
1066/12= 88.8 years
Find all real zeros of the function y = -7x + 8
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Answer:
x = 8/7
Step-by-step explanation:
The only real zero of this linear function is the value of x that makes y=0:
0 = -7x +8
7x = 8 . . . . . . add 7x
x = 8/7 . . . . . .divide by 7
The endpoints of DEF are D(1, 4) and F(16, 14).
Determine and state the coordinates of point E, if
DE: EF = 2:3.
Answer:
The coordinates of point E are (7,8).
Step-by-step explanation:
Point E:
Is given by (x,y).
DE: EF = 2:3.
This means that, for both coordinates x and y:
[tex]E - D = \frac{2}{2+3}(F-D)[/tex]
[tex]E - D = \frac{2}{5}(F-D)[/tex]
x-coordinate:
x-coordinate of D: 1
x-coordinate of F: 16
[tex]E - D = \frac{2}{5}(F-D)[/tex]
[tex]x - 1 = \frac{2}{5}(16-1)[/tex]
[tex]x - 1 = 2*3[/tex]
[tex]x = 7[/tex]
y-coordiante:
y-coordinate of D: 4
y-coordinate of F: 14
[tex]E - D = \frac{2}{5}(F-D)[/tex]
[tex]y - 4 = \frac{2}{5}(14-4)[/tex]
[tex]y - 4 = 2*2[/tex]
[tex]x = 8[/tex]
The coordinates of point E are (7,8).
Function limits: Solve the activities in each image (full development)
Answer with the method that is possible to solve adequately (approximation, factorization, rationalization or evaluation)
Step-by-step explanation:
the answer is in the above image
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Answer:
1a. f(4) = 1/2
1b. DNE
2a. f(2) = 5
2b. f(-3/2) = 9 1/4
2c. 4
2d. -2
2e. -1/3
Step-by-step explanation:
The limit for functions that can be evaluated is simply the value of the function. (1a, 2a, 2b are like that)
Some rational functions, particularly the ratios of polynomials, can be simplified by cancelling common factors from numerator and denominator. The simplified form can often be evaluated directly at the variable value that leaves the original function "undefined." (2c and 2d are like that)
Where a (simplified) rational function has an odd-degree zero in the denominator, there is no limit at that point. The function value changes sign at the discontinuity, so the left limit is different from the right limit. The limit Does Not Exist. (1b is like that)
Indeterminate forms (0/0 or ∞/∞)have their own methods for finding limits. Chief among these is L'Hopital's rule, which has you compare derivatives of numerator and denominator, repeating as necessary if those also give an indeterminate form. I like to gain a clue from the graph. Often, the function can be evaluated "arbitrarily close" to the limiting value of the variable, so the limit can be guessed at without much trouble.
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1a. Evaluate the function: 1/(4-2) = 1/2
1b. the denominator changes sign at x=2, so the limit Does Not Exist
2a. Evaluate the function: (2-2)(2) +5 = 5
2b. Evaluate the function: (-3/2 -4)(-3/2) +1 = 33/4 +1 = 37/4 = 9 1/4
2c. Simplify and evaluate: (x -3)(x -7)/(x -7) = x -3 ⇒ 7 -3 = 4
2d. Simplify and evaluate: (x +1)(x +3)/(x +3) = x +1 ⇒ -3 +1 = -2
2e. The function has a hole at x=4 but can be evaluated nearby (approximation). (See attached). The limit is -1/3. (The ratio of derivatives per L'Hopital's rule reduces to -(√5-x)/√(5+x) = -√(1/9) = -1/3.)
