1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Maksim231197 [3]
3 years ago
12

Can someone help me with this? I need to find the points of discontinuity/limits for each of these. I think one point is 4, but

I'm not sure about the others.

Mathematics
1 answer:
Debora [2.8K]3 years ago
6 0
The answers are shown in the attached image

-------------------------------------------------------------------------

Explanation:

Set the denominator x^4-8x^3+16x^2 equal to zero and solve for x

x^4-8x^3+16x^2 = 0
x^2(x^2-8x+16) = 0
x^2(x-4)^2 = 0
x^2 = 0 or (x-4)^2 = 0
x = 0 or x-4 = 0
x = 0 or x = 4

The x values 0 and 4 make the denominator zero

These x values lead to asymptote discontinuities because the numerator 8x-24 = 8(x-3) has no common factors which cancel with the denominator factors.

There are two vertical asymptotes

Let's see what happens when we plug in a value to the left of x = 0, say x = -1, we'd get
f(x) = (8x-24)/(x^4-8x^3+16x^2)
f(-1) = (8(-1)-24)/((-1)^4-8(-1)^3+16(-1)^2)
f(-1) = -1.28
So as x gets closer and closer to x = 0 from the left side, the f(x) is heading to negative infinity

Now plug in some value to the right of x = 0. I'm going to pick x = 1
f(x) = (8x-24)/(x^4-8x^3+16x^2)
f(1) = (8(1)-24)/((1)^4-8(1)^3+16(1)^2)
f(1) = -1.78 (approximate)
So as x gets closer and closer to x = 0 from the right side, the f(x) is heading to negative infinity

Overall, as x approaches 0 from either the left or right side of x = 0, the y value is heading off to negative infinity

---------------------

Repeat for values to the left and right of x = 4
We can't use x = 1 as it turns out that x = 3 is a root
But we can use something like x = 3.5 to find that...
f(x) = (8x-24)/(x^4-8x^3+16x^2)
f(3.5) = (8(3.5)-24)/((3.5)^4-8(3.5)^3+16(3.5)^2)
f(3.5) = 1.31 approx
So as x gets closer to x = 4 from the left, y is getting closer to positive infinity

Plug in x = 5 to find that
f(x) = (8x-24)/(x^4-8x^3+16x^2)
f(5) = (8(5)-24)/((5)^4-8(5)^3+16(5)^2)
f(5) = 0.64
which has the same behavior as the left side

So overall, as we approach x = 4, the y value is heading off to positive infinity

Again everything is summarized in the image attachment

Note: you could make a table of more values but they would effectively say what has already been said. It would be redundant busy work. However, its always good practice for function evaluation. 

You might be interested in
The base of the pyramid is a square what is the surface area if the pyramids height is 4 feet in the area of the base is 36 ft.?
KATRIN_1 [288]

Answer:

96 ft²

Step-by-step explanation:

The general formula for the surface area of a pyramid is:  

SA = \frac{1}{2}pl + B, were p = the perimeter of the base, l = the slant height and B = the area of the base

Since the area of the base is 36 ft², then each side is √36, or 6 ft. The perimeter is the sum of all the sides, so 6 x 4 = 24 ft.

Given the overall height of the pyramid is 4 feet, you can use the Pythagorean Theorem (a² + b² = c²) to find the slant height (hypotenuse) of the pyramid.  Using 3 feet for 'a' and 4 feet for 'b':

3² + 4² = c² or 9 + 16 = 25

25 = c² or c = 5

SA =  \frac{1}{2}(24)(5) + 36

SA = 60 + 36 = 96 ft²

8 0
3 years ago
Order these numbers from least to greatest. 80 , −9.75 , −9.7 , −24825
Diano4ka-milaya [45]
Least is on top, Greatest is on bottom
-24,825
-9.75
-9.7
80
8 0
3 years ago
The __________ set is the set of elements under consideration
dedylja [7]

Answer:

SYNTAX PROPERTY

Step-by-step explanation:

4 0
3 years ago
One bag of rice that weighs 5.25 pounds will have enough rice if a recipe calls for 88 ounces of rice. (1 pound = 16 ounces)
krek1111 [17]
False, there isn't enough rice in 5.25 pounds.
4 0
3 years ago
If a puppet requires 3/4 yard of material, how many puppets can be made from 9 yards of material?
klasskru [66]
First find out how many times 3/4 yard goes into 9. 3/4 fits into 9 twelve times, so twelve puppets can be made.

Did I help? Thank me by clicking best answer!
5 0
3 years ago
Read 2 more answers
Other questions:
  • A regular octagon rotates 360° about its center. How many times does the image of the octagon coincide with the preimage during
    7·1 answer
  • 10. If AB = BC, AX = 4x + 5, and<br> CX = 15, find AC.
    6·1 answer
  • 1. In the number 7563291 write down the column that the 5 is in
    6·1 answer
  • 1. Ada purchased a house using a fixed rate mortgage. The annual interest rate is 6.3%
    7·1 answer
  • Please answer this 1/24x2=1/12x2-1/x
    14·1 answer
  • Theo bought 8 packs of pencils for $4 each and 3 packs of
    12·1 answer
  • Tony writes an equation to represent the distance he walks in meters, d, in relation to the number of minutes he walks, m.
    8·2 answers
  • Please help! Thank you!!
    9·2 answers
  • Jim is on his way home in his car. He has driven 20 miles so far which is 1/2 of the way home. What is the total length of his D
    5·1 answer
  • Random questions
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!