Limits
Asymptotes
Discontinuity
End Behavior
Miscellaneous
100
The limit of sin x as x approaches infinity is A. 1 B. -1 C. 0 D. Limit does not exist
D. Limit does not exist. Sin x goes on forever bouncing between y=1 and y=-1.
100
What are the vertical and horizontal asymptotes (if any) of f(x) = (4)/((x^2)-1)?
The x axis acts as a horizontal asymptote and x=±1 are the vertical asymptotes.
100
Where is the function f(x)=(x^2-4)/(x-2) discontinuous?
What is x=2?
100
Describe the limits as x approaches ∞ and -∞. f(x)= x^3-3x²+7x+2.
Limit as x approaches ∞ is ∞. Limit as x approaches -∞ is -∞.
100

What is the limit of sin (Θ)/ Θ as Θ approaches x?

sin(x)/x

200
What is the limit of x/x as x approaches 0?
1. The fraction equals 1 for all non-zero x values.
200
How many vertical and horizontal asymptotes (if any) does the graph of y=(2x^2+2x+3)/(4x^2-4x) have?
It has one horizontal asymptote at y=1/2 and two vertical asymptotes at x=0 and x=1.
200
Is f(x)=(x^2+2) x<1, and (4) x>1 continuous at x=1?
No it is not continuous at x=1 because as x approaches 1 from the left the limit is 3 and from the right the limit is 4.
200
Describe the limits as x approaches ∞ and -∞. g(x)=-4x^4+1,000,000x^3+100
The limit as x approaches ∞ is -∞. The limit as x approaches -∞ is also -∞.
200
What is the limit of sin x /x as x approaches ∞?
0
300

What is the limit as x approaches 2 of 

(x^2 + 2x - 8)/ (x^2 -4)?

3/2. Take out an (x-2) from both the top and bottom to help solve.

300
What vertical and horizontal asymptotes (if any) does the graph of y=(x^2-9)/(3x-9) have?
There are no horizontal or vertical asymptotes, but the function is has a removable discontinuity at x=3.
300

For a function f(x)=x^2 when x is not =2, and f(x) = 1 when x=2. Is f(x) continuous at x = 2?

No, the limit as x approaches 2 is 4 and the value is 2. Therefore, this is not continuous.

300
Describe the limits as x approaches ∞ and -∞. h(x)= -5x^3 + 3x² - 4π +8.
The limit as x approaches ∞ is -∞ and the limit as x approaches -∞ is ∞.
300

Rewrite the following:

lim as x approaches 0 of f(x)*g(x)

(lim as x approaches 0 of f(x))*(lim as x approaches 0 of g(x))

400
What is the limit of (sin 2x)/3x as x approaches 0?
2/3. Begin by pulling out 1/3 from the original equation. Then multiply both the top and bottom by 2. This will give you 2/3 lim sin 2x/(2x).
400
Where do the asymptotes occur on the graph of y=arctanx?
They occur at y=±π/2. We know this because in they occur at x=±π/2 on the graph of the regular tan x function.
400
Determine if the following function is continuous at x=-2. f(x)= {x²+2x if x≤-2, x^3-6x if x≥-2.
The left hand limit is 0 and the right hand limit is 4. Since the left and right hand limits are not equal, a limit does not exist at x=-2.
400
Describe the limits as x approaches ∞ and -∞. k(x)= π-.001x
The limit as x approaches ∞ is -∞. The limit as x approaches -∞ is ∞.
400

Simplify the following: lim as x approaches k of x is equal to...

K. The limit is approaching k.

500
What is the limit of sin x/(x^2+3x) as x approaches 0?
1/3. Begin by pulling out an x from the bottom half of the equation. Then separate the equation into sin x/(x) times 1/(x+3).
500
Find the horizontal asymptote(s) of the following function: y=(8x²+3x+4)/(2x²+99).
Both polynomials are to the second degree, so the asymptote is at y=8/2 or y=4.
500
For what x values is the function (x²+3x+5)/(x²+3x-4) continuous?
The function is continuous on all values except x=1 and x=-4.
500
Describe the limits as x approaches ∞ and -∞. f(x)= x^3-4x^2+7.
Limit as x approaches ∞ is ∞. Limit as x approaches -∞ is -∞.
500

Rewrite the following: lim [f(x) + g(x)]

lim f(x) + lim g(x)

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