AP Calculus AB : Limits of Functions (including one-sided limits)
Study concepts, example questions & explanations for AP Calculus AB
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Example Questions
Example Question #13 : Understanding The Limiting Process.
Find the derivative of:
This problem involves the chain rule for derivatives. However, you must first rewrite the function as:
Then, apply the chain rule (first multiply the exponent by the coefficient in front of the binomial [1], then decrease the exponent of the binomial by 1, and finally take the derivative of the binomial):
When simplifiying, change negative exponents to positive ones. Therefore, the answer is:
Example Question #15 : Understanding The Limiting Process.
If 
The correct answer is 
We must use the product rule to solve. Remember that the derivative of 
Example Question #51 : Limits Of Functions (Including One Sided Limits)
Differentiate 
The derivative of 
The second part requires regular differential rules.
Therefore when differentiating 
Combining the first and second part we get the final derivative:
Example Question #52 : Limits Of Functions (Including One Sided Limits)
Evaluate:
The derivative of 
Therefore the integral of
Example Question #52 : Limits Of Functions (Including One Sided Limits)
Differentiate 
The rule for taking the derivative of 
For this problem we need to remember to use the Chain Rule.
Since we are taking the derivative of,
Therefore the solution becomes:
Example Question #53 : Limits Of Functions (Including One Sided Limits)
If 
The derivative of,
The derivative of
Therefore, using the Chain Rule the derivative of the function will become the derivative of the outside piece keeping the original inside piece. Then multiplying that by the derivative of the inside piece.
Example Question #16 : Understanding The Limiting Process.
Evaluate the following limit:
does not exist
Recall the formal definition of the derivative:
When you evaluate this limit the output is f'(x). In this question f(x) = ln(x) so what this question is really saying is take the derivative of f(x) and evaluate it at 2.
Take the derivative:
Substitute the value 2 into the derivative:
Example Question #53 : Limits Of Functions (Including One Sided Limits)
Evaluate the following limit:
This limit is very simple(almost too simple) because it asks for the limit at a location where there is no discontinuity. Fortunately, this makes taking the limit trivial.
Substitute x=4 into the function to evaluate the limit.
Example Question #18 : Understanding The Limiting Process.
Evaluate the following limit:
As x becomes infinitely large, the x terms with the highest power dominate the function and the terms of lower order become negligible. This means that near infinity, the x^5 term in the numerator and the 10X^5 term in the denominator are the only values necessary to evaluate the limit.
Simplify and evaluate:
Example Question #19 : Understanding The Limiting Process.
Evaluate the following limit:
does not exist
does not exist
First, factor x-2 out from the numerator and denominator.
At x=4 there is a discontinuity, so we must evaluate the limit from the right and left side to see if it exists.
Evaluated from the right:
From the left:
Because the limit from the right is not equal to the limit from the left, the function does not exist at x=4.
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