Answers
Answer:
Yes, the limit does exist.
Step-by-step explanation:
By definition, a limit in the form:
[tex]\displaystyle \lim_{x\to a}f(x)[/tex]
Exists if and only if:
[tex]\displaystyle \lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)[/tex]
We are given the limit:
[tex]\displaystyle \lim_{x\to 2}\frac{x^2+x-6}{x^2-4}[/tex]
So, in order to prove that this limit exists, we simply have to show that:
[tex]\displaystyle \lim_{x\to 2^-}\frac{x^2+x-6}{x^2-4}=\displaystyle \lim_{x\to 2^+}\frac{x^2+x-6}{x^2-4}[/tex]
Let's do the left-hand side first. We can factor the function:
[tex]=\displaystyle \lim_{x\to 2^-}\frac{(x+3)(x-2)}{(x+2)(x-2)}[/tex]
Simplify:
[tex]\displaystyle =\lim_{x\to 2^-}\frac{x+3}{x+2}[/tex]
By direct substitution:
[tex]\displaystyle =\frac{(2)+3}{(2)+2}=\frac{5}{4}[/tex]
Now, we can evaluate the right-hand side. Again, factor:
[tex]=\displaystyle \lim_{x\to 2^+}\frac{(x+3)(x-2)}{(x+2)(x-2)}[/tex]
Simplify:
[tex]\displaystyle =\lim_{x\to 2^+}\frac{x+3}{x+2}[/tex]
And by direct substitution:
[tex]\displaystyle =\frac{(2)+3}{(2)+2}=\frac{5}{4}[/tex]
Since both the left- and right-hand limits equal 5/4, our original limit does indeed exist.
Related Questions
WILL GIVE BRANLIEST.
Which of the following statements are true?
Select 4 correct answers
1. All of the coordinate pairs in Quadrant 1 are positive.
2. All of the coordinate pairs in Quadrant 3 are negative.
3 All of the coordinate pairs in Quadrant 2 are positive.
4 Everything to the left of zero on the x axis is negative, everything below zero on the y axis is negative.
5The x axis is positive in quadrants 1 & 3
6 The y axis is negative in Quadrants 3 & Quadrant 4.
Answers
Answer:
2,3,4, and 5
Step-by-step explanation:
#1
In Q1 x>0,y>0So all are positive
#2
In Q3
x<0y<0Hence all are negative
#4
Yes anything left to or bottom to zero us negative
#6
Yes on Both Q3 and Q4 y<0option A,B,D,F are correct
Jemima has twice as much money
as Matthew. Jemima has four times
as much money as Sally, Sally has
$3 more than Andrew. If Matthew
has $14, how much money do
Andrew, Sally and Jemima have?
Answers
Sally has $7
Jemima has $28
Jemina has twice as much as Matthew
Mathew has $14
14•2=28
Jemima has $28
Jemima has four times as much as sally
28/4=7
Sally has $7
Sally has $3 more than Andrew
7-3=4
78% of all laboratory mice can make it through a maze. If 600 randomly selected mice attempt the maze. Write out the formula to calculate the confidence interval
Answers
Explanation:
i have to finish this today HELPPPP!
a teacher asks five students to write an equation with a solution of x= 4 . the students and their equations are shown below. which of the following students correctly wrote an equation with x= 4 as the solution ? select three that apply
Answers
Bert, Camille, and Etsuko.
Answer:
Bert, Camille, and Etsuko has the solution as x = 4.
Step-by-step explanation:
Bert:
2x - 9 = -1
Camille:
-2x + 12 = 4
Etsuko
[tex]2(x - 10) = - 12[/tex]
At WMS, 3 out of 4 students ride a bus to school. There are 600 students at WMS. How many students ride the bus?
Answers
Answer:
450 students
Step-by-step explanation:
Your grocery store bill was $34.97. You paid with a $50 bill. How much money do you have left?
Answers
Answer:
$15.03
Step-by-step explanation:
50.00
- 34.97
_______
15.03
Answer:
The answer is $15.03.
Step-by-step explanation:
Just subtract.
(PLSSSS HELP) Determine which three out of the six points shown below are a solution to the equation x-3y=22 Plot the three points that are in the solution set, draw a line through the three points and then answer the questions below.
Answers
Answer:
[tex](-2,8)[/tex] [tex](7,5)[/tex] [tex](1,7)[/tex]
Step-by-step explanation:
Given
[tex]x + 3y = 22[/tex]
Solving (a): The possible coordinates
We have:
[tex](x,y) = (-2,8)[/tex]
This gives:
[tex]x + 3y = 22 \to -2 + 3 * 8[/tex]
[tex]x + 3y = 22 \to -2 + 24[/tex]
[tex]x + 3y = 22 \to 22[/tex]
(-2,8) is a point on the grid
[tex](x,y) = (7,5)[/tex]
This gives:
[tex]x + 3y = 22 \to 7 + 3 * 5[/tex]
[tex]x + 3y = 22 \to 7 + 15[/tex]
[tex]x + 3y = 22 \to 22[/tex]
(7,5) is a point on the grid
[tex](x,y) = (-5,5)[/tex]
This gives:
[tex]x + 3y = 22 \to -5 + 3 * 5[/tex]
[tex]x + 3y = 22 \to -5 + 15[/tex]
[tex]x + 3y = 22 \to 10[/tex]
(-5,5) is a not point on the grid
[tex](x,y) = (1,3)[/tex]
This gives:
[tex]x + 3y = 22 \to 1 + 3 * 3[/tex]
[tex]x + 3y = 22 \to 1 + 9[/tex]
[tex]x + 3y = 22 \to 10[/tex]
(1,3) is a not point on the grid
[tex](x,y) = (1,7)[/tex]
This gives:
[tex]x + 3y = 22 \to 1 + 3 * 7[/tex]
[tex]x + 3y = 22 \to 1 + 21[/tex]
[tex]x + 3y = 22 \to 22[/tex]
(1,7) is a point on the grid
So, we plot the following on the grid
[tex](-2,8)[/tex] [tex](7,5)[/tex] [tex](1,7)[/tex]
See attachment
Why Was A Book In The Frying Pan On The Stove?
Answers
Pls help i'll give brainlest
Answers
Answer: B. 183 cm^3
V = pi r^2 h/3
V = pi (5^2) (7/3)
V = pi (25) (2.33)
V = 183
Answer:
B.183 [tex]cm^{3}[/tex]
Step-by-step explanation:
Volume of cone is [tex]V=\pi r^{2} \frac{h}{3}[/tex]
where h=7 and r=5 (because r=d/2, where d=10)
V=[tex]5^{2}\frac{7}{3}\pi[/tex]
V=25·[tex]\frac{7}{3} \pi[/tex]
V=175/3[tex]\pi[/tex]
V=183.2595715 or 183 [tex]cm^{3}[/tex]
The cardinal number of {17, 18, 19, 20, ..., 95} is
Answers
Answer:
95
Step-by-step explanation:
a cardinal number is anything that is to 70 and the rest of the numbers above 70 are counting by 10s
Solve this system of equations using the substitution method y=5x+4 x+12=y
Answers
Answer:
(2,14)
Step-by-step explanation:
Since y=5x+4, you can plug that into the other y in the other equation which would give you x+12=5x+4
Now subtract 12 from both sides
x=5x-8
Subtract 5x from both sides
-4x=-8
Now you divide them both by -4 which gives you x=2
Now plug x in to any of the equations
y=5(2)+4
y=10+4
y=14
Your final answer is (2,14)
Why is c speed of light in albert einseins eqation e=mc squared
Answers
Answer:
[tex]{\huge\pink{\fbox{{࿐αɴѕωєя࿐}}}}[/tex]
The speed of light (c) is considered a fundamental constant of nature. In the famous relativity equation, E = mc², the speed of light (c) serves as a constant of proportionality, linking the formerly disparate concepts of mass (m) and energy (E).
Ssomeone help me in this one now please
Answers
You deposited 750 in an account that pays 5% interest compounded quarterly. How many years do you have two wait until it becomes at least ten times as much? Round answer to the nearest whole year.
A. 47
B. 46
C. 45
D. 48
Answers
Question 4 of 10
When the input is x = 3 for the parent function f(x) = x, what is the output?
O A. f(x) = 3
O B. f(x) = -3
O C. f(x) = 0
O D. f(x) = 1
Answers
Answer:
A
Step-by-step explanation:
f(x) = x
x = 3
->
f(3) = 3
A
Hans is performing an experiment examining how salt affects the speed at which water freezes. He fills four equally sized containers with the same amounts of water. Then, in three of the containers, he stirs in varying amounts of salt. The fourth container remains plain water.
Hans places all four of the containers in the same freezer and checks the containers every five minutes, recording when the water in each container freezes solid.
In this situation, what do Hans’s time measurements represent?
A.
the response variable
B.
the control group
C.
the response group
D.
the explanatory variable
Answers
Answer:
Answer B.
Because we know He's checking the experiment every 5 minutes.
Find the missing length indicated
A) 25
B) 5
C) 8
D) 3
Answers
Given:
The figure of a triangle.
To find:
The missing length.
Solution:
Let x be the missing length.
Basic proportionality theorem(BPT): According to this theorem, if a line is parallel to the one side of a triangle then that the line intersects the other two sides of the triangle in the same ratio.
Using Basic proportionality theorem, we get
[tex]\dfrac{6}{8-6}=\dfrac{12-x}{x}[/tex]
[tex]\dfrac{6}{2}=\dfrac{12-x}{x}[/tex]
[tex]3=\dfrac{12-x}{x}[/tex]
[tex]3x=12-x[/tex]
Adding x on both sides, we get
[tex]3x+x=12[/tex]
[tex]4x=12[/tex]
[tex]x=\dfrac{12}{4}[/tex]
[tex]x=3[/tex]
The missing length is 3 Therefore, the correct option is D.
A pizza delivery chain advertises that it will deliver your pizza in 30 minutes from when the order is placed. Being a skeptic, you decide to test and see if the mean delivery time is actually more than 30 minutes. For the simple random sample of 9 customers who record the amount of time it takes for each of their pizzas to be delivered, the mean is 35.4 minutes with a standard deviation of 7.6 minutes. Assume that the population distribution is approximately normal.
Required:
a. Perform a hypothesis test using a 0.05 level of significance.
b. State the null and alternative hypotheses for the test.
Answers
Answer: B
Step-by-step explanation: So that you know you are only getting the best for what YOU PAID for.
translate the point (5,2)—> 3 units to the left
Answers
Given:
The point (5,-2) is translated 3 units to the left.
To find:
The new location of the point.
Solution:
If a point is translated 3 units to the left, then
[tex](x,y)\to (x-3,y)[/tex]
Using this rule of translation, we get
[tex](5,-2)\to (5-3,-2)[/tex]
[tex](5,-2)\to (2,-2)[/tex]
Therefore, the new location of the point is (2,-2). Hence, the correct option is C.
Find the coefficient of the x^3 in the expansion of (2x-9)^5
Answers
Use the binomial theorem:
[tex]\displaystyle (2x-9)^5 = \sum_{k=0}^5 \binom5k (2x)^{5-k}(-9)^k = \sum_{k=0}^5 \frac{5!}{k!(5-k)!} 2^5 \left(-\frac92\right)^k x^{5-k}[/tex]
The x ³ terms occurs for 5 - k = 3, or k = 2, and its coefficient would be
[tex]\dfrac{5!}{2!(5-2)!} 2^5 \left(-\dfrac92\right)^2 = \boxed{6480}[/tex]
Reece is setting up a large fish tank. The tank is 1.8metres long and 0.6 metres wide. The water in the tank is 50 cm deep. Reece needs to add a liquid treatment to the water. The instructions tell him to add 5 ml of treatment for every 0.18 m3. How many treatment does Reece need to add?
Answers
Answer:
15 ml
Step-by-step explanation:
Find the volume of the tank which is 1.8m*0.6m*0.5m=0.54m^3
For every 0.18m^3,5ml of treatment is added,what of 0.54m^3
(0.54m^3*5ml)/(0 18m^3)=15ml
combination
in a lottery, a player chooses 6 different numbers from 1-40 and if they match the winning numbers the player wins the top prize
what is the probability of winning the top prize?
Answers
Answer:
I would maybe say 34% im in sixth grade and this hard for me atleast i tried
Step-by-step explanation:
Answer:
1 out of 27,636,336,000
Step-by-step explanation:
40 · 39 · 38 · 37 · 36 · 35 = 27,636,336,000
While riding her bike to the ocean and back home 6 times, Melissa timed herself at 42min, 46min, 52min, 57min, 59min, and 55min. She had set a goal of averaging 51 minutes for her rides. How many minutes will she need to spend on her seventh ride to attain her goal?
Answers
Answer:
She needs a time of 61 min on her 8th ride
in order to average 51 min for the 8 rides
Step-by-step explanation:
+%28+46+%2B+48+%2B+50+%2B+43+%2B+54+%2B+54+%2B+52+%2B+t+%29+%2F+8+=+51+
+347+%2B+t+%29+%2F+8+=+51+
+347+%2B+t+=+408+
+t+=+61+
Help me please!
Consider the function f(x) = V2x – 4. Iff-'(x) is the inverse function of f(x), find
f-16)
Answers
Answer:
[tex]f^{-1}(6) = 50[/tex]
Step-by-step explanation:
Given
[tex]f(x) = \sqrt{2x} - 4[/tex]
Required
Find [tex]f^{-1}(6)[/tex]
First, we calculate the inverse function
[tex]f(x) = \sqrt{2x} - 4[/tex]
Express f(x) as y
[tex]y = \sqrt{2x} - 4[/tex]
Swap the positions of x and y
[tex]x = \sqrt{2y} - 4[/tex]
Solve for y: Add 4 to both sides
[tex]4 + x = \sqrt{2y} - 4+4[/tex]
[tex]4 + x = \sqrt{2y}[/tex]
Square both sides
[tex](4 + x)^2 = 2y[/tex]
Divide both sides by 2
[tex]y = \frac{(4 + x)^2}{2}[/tex]
Express y as an inverse function
[tex]f^{-1}(x) = \frac{(4 + x)^2}{2}[/tex]
Next, solve for: [tex]f^{-1}(6)[/tex]
Substitute 6 for x
[tex]f^{-1}(6) = \frac{(4 + 6)^2}{2}[/tex]
[tex]f^{-1}(6) = \frac{(10)^2}{2}[/tex]
[tex]f^{-1}(6) = \frac{100}{2}[/tex]
[tex]f^{-1}(6) = 50[/tex]
Miguel is 4 years older than 2/3 of Harper's age. The sum of their ages is 34. His harper 16?
Answers
Answer:
Harper's age = 18
Miguel age = 16
Step-by-step explanation:
Given:
Miguel age = [2/3][Harper's age] + 4
Sum of their age = 34
Computation:
Assume;
Harper's age = a
So,
Miguel age = [2/3][a] + 4
Sum of their age = 34
So,
Miguel age + Harper's age = 34
[2/3][a] + 4 + a = 34
[2a / 3] + a = 34 - 4
[2a + 3a] / 3 = 30
5a / 3 = 30
5a = 90
a = 18
So,
Harper's age = 18
Miguel age = [2/3][a] + 4
Miguel age = 16
The volume of a sphere is 113.04 cubic feet. Find the radius. Use 3.14 for pi.
Helppppppp pls
Answers
the radius is 5.2 ft
A shopkeeper purchased 60 pencil boxes for 4.500 and sold at the rate of 4.125 for 50 pencil boxes. How
much profit present does he make on selling all the pencil boxes?
5%
7%
10%
12%
Answers
Answer: 10%
Step-by-step explanation:
Since the shopkeeper purchased 60 pencil boxes for 4,500, he purchased them at the rate of:
= 4500 / 60
= 85 per pencil box
If the shopkeeper sold them at the rate of 4,125 for 50 pencil boxes, then he sold them at the rate of:
= 4125 / 50
= 82.50 per pencil box.
Therefore the percentage profit will be:
= Gain / Cost price × 100
= (82.50 - 75) / 75 × 100
= 7.5/75 × 100
= 10%
Are all proportional relationships functions?
Answers
What is the domain? Is it (-4,3)
Answers
Yes I think you are correct. The domain appears to be the set of x values such that -4 < x < 3 meaning that x is between -4 and 3, excluding each endpoint. In interval notation, that translates to (-4, 3). This of course assumes that we have two vertical asymptotes bookending the left and right sides of the curve, and these vertical asymptotes are at x = -4 and x = 3. If the curve extends beyond these boundaries, then the domain would instead be the set of all real numbers.
Pls answer. I will mark brainliest.
Answers
Answer:
true
false
true
false
true
false
Step-by-step explanation: