AP Calculus AB : Functions, Graphs, and Limits
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Example Questions
Example Question #331 : Ap Calculus Ab
Evaluate:
For this problem we need to use the U Substitution Method.
Using the U-du Rule you can set 
Because we only have a dx in our problem we need to solve for dx, thus
When 
Therefore your new equation will be:
Plugging in our interval we get,
Example Question #125 : Asymptotic And Unbounded Behavior
Evaluate:
The first step is to find the antiderivative, recalling that:
For this integral:
where the intergral would be evaluated from 
Example Question #126 : Asymptotic And Unbounded Behavior
Evaluate the following indefinite integral:
Use substitution, where 
Substitution of 
Note that since this is an indefinite integral, the addition of a constant term (C) is required.
Example Question #131 : Functions, Graphs, And Limits
Evaluate the limit:
The limit does not exist
When evaluating the limit as x approaches infinity, we must compare the magnitude of the functions. The exponential function in the numerator grows faster than the polynomial function (and any polynomial, for that matter) in the denominator, so the numerator dominates and the limit equals 
Example Question #1 : Calculating Limits Using Algebra
Calculate
5
10
The limit does not exist
0
10
First we notice that substituting 5 in for x will give us a 0 in the denominator.
So we simplify the equation by noticing the numerator is the difference of two squares.
Now we can substitute 5 in for x, and we arrive at our answer of 10.
Example Question #2 : Calculating Limits Using Algebra
Evaluate the following limit:
Factor x-4 out of the numerator and simplify:
Evaluate the limit for x=4:
Although there is a discontinuity at x=4, the limit at x=4 is 10 because the function approaches ten from the left and right side.
Example Question #3 : Calculating Limits Using Algebra
Evaluate the following limit:
Factor the numerator and simplify the expression.
Evaluate the function at x=2.
There is a discontinuity at x=2, but since it the limit as x approaches 2 from the right is equal to the limit as x approaches 2 from the left, the limit exists.
Example Question #4 : Calculating Limits Using Algebra
Evaluate the following limit:
does not exist
Factor the numerator to evaluate the limit:
Evaluate the limit:
There is a discontinuity at x=0 but the limit is equal to 8 because the limit from the right is equal to the limit from the left.
Example Question #5 : Calculating Limits Using Algebra
Evaluate the following limit:
As x approaches infinity, the x values that are raised to the highest power dominate the function and all other values become negligible. In this limit, x3 has the highest order in the numerator and 5x2 has the highest power of all the values in the denominator. We can simplify the limit to the following:
After factoring out x2 it is easy to see that the limit diverges to infinity.
Example Question #1 : Limits Of Functions (Including One Sided Limits)
Undefined.
The first step to evaluating limits is always attempt to direct substitute the value into the function. In this case, that is all you need to do.
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