Limits
What are the vertical and horizontal asymptotes (if any) of f(x) = (4)/((x2)-1)?
The x axis acts as a horizontal asymptote and x=±1 are the vertical asymptotes.
Where is the function f(x)=(x2-4)/(x-2) discontinuous?
What is x=2?
Describe the limits as x approaches ∞ and -∞. f(x)= x3-3x²+7x+2.
Limit as x approaches ∞ is ∞. Limit as x approaches -∞ is -∞.
What is the limit of sin Θ/ Θ as Θ approaches x?
1 if Θ is measured in radians.
How many vertical and horizontal asymptotes (if any) does the graph of y=(2x2+2x+3)/(4x2-4x) have?
It has one horizontal asymptote at y=1/2 and two vertical asymptotes at x=0 and x=1.
Is f(x)=(x2+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.
Describe the limits as x approaches ∞ and -∞. g(x)=-4x4+1,000,000x3+100
The limit as x approaches ∞ is -∞. The limit as x approaches -∞ is also -∞.
What is the limit of (x3 - 8)/ (x2 -4) as x approaches 2?
1. Take out an (x-2) from both the top and bottom to help solve.
What vertical and horizontal asymptotes (if any) does the graph of y=(x2-9)/(3x-9) have?
There are no horizontal or vertical asymptotes, but the function is has a removable discontinuity at x=3.
Is f(x)=x2 x is not =2, and 1 x=2, continuous at x=2?
No this is a removable discontinuity at x=2.
Describe the limits as x approaches ∞ and -∞. h(x)= -5x3 + 3x² - 4π +8.
The limit as x approaches ∞ is -∞ and the limit as x approaches -∞ is ∞.
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).
Where do the asymptotes occur on the graph of y=tanx?
They occur at x=±pi/2 on the graph of the regular tan x function.
Determine if the following piecewise function is continuous at x=-2. f(x)= {x²+2x if x≤-2 and x3-6x 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.
What is the limit of sin x/(x2+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).
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.
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.
Describe the limits as x approaches ∞ and -∞. f(x)= x3-4x2+7.
Limit as x approaches ∞ is ∞. Limit as x approaches -∞ is -∞.