Definition
definition of slope
rate of change
What is the limit of sin Θ/ Θ as Θ approaches x?
1 if Θ is measured in radians.
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.
What is when the finite left-sided limit does not equal to the finite right-sided limit.
The instance when there is a jump discontinuity at x = c.
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.
definition of secant line
the slope of the line through these two point.
What is the limit of sin x /x as x approaches ∞？
0
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.
What is removable discontinuity?
The type of discontinuity in which both sided limit approaching "c" exists, but is not equal to f(c), which is defined.
What is the limit of x/x as x approaches 0?
The fraction equals 1 for all non-zero x values.
Definition of tangent line
hits the curve at 1 point.
Simplify the following: lim f(x) g(x)
(lim f(x)) (lim g(x))
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.
What is a jump discontinuity at x = c ?
The discontinuity when the finite left-sided limit does not equal to the finite right-sided limit as x approaches "c". The limit from both sides is not existent.
What is the limit of (x^3 - 8)/ (x^2 -4)?
Take out an (x-2) from both the top and bottom to help solve.
What are 3 situations, the graph is discontinuous?
1. Vertical Asymptote
2. Jump=piecewise function
3. hole
Simplify the following: lim k as x approaches k
K. The limit is approaching k.
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.
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.
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).
What is the 6 situations, the graph is no derivatives?
1. Vertical Asymptote
2. Jump=piecewise function
3. hole
4. vertical line=no slope
5. cusp
6. absolute value
Simplify the following: lim [f(x) + g(x)]
lim f(x) + lim g(x)
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.
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.
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).