# AP Calculus AB : Functions, Graphs, and Limits

## Example Questions

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### Example Question #1 : Geometric Understanding Of Graphs Of Continuous Functions

is differentiable for which of the following values of

Explanation:

is not differentiable at  and  because the values are discontinuities.  is not differentiable at  because that point is a corner, indicating that the one-side limits at  are different.  is differentiable:the one side limits are the same and the point is continuous.

### Example Question #2 : Geometric Understanding Of Graphs Of Continuous Functions

Consider the graph of  above. What can we say about  when  ?

has a horizontal tangent at .

has a removable discontinuity at .

is discontinuous at  because there is a sharp turn at .

Two or more of these are correct.

None of these are correct.

has a horizontal tangent at .

Explanation:

Note that , indicating that there is a horizontal tangent on  at . More specifically, the derivative is the slope of the tangent line. If the slope of the tangent line is 0, then the tangent is horizontal.

The other two are incorrect because sharp turns only apply when we want to take the derivative of something. The derivative of a function at a sharp turn is undefined, meaning the graph of the derivative will be discontinuous at the sharp turn. (To see why, ask yourself if the slope at  is positive 1 or negative 1?) On the other hand, integration is less picky than differentiation: We do not need a smooth function to take an integral.

In this case, to get from  to , we took an integral, so it didn't matter that there was a sharp turn at the specified point. Thus, neither function had any discontinuities.

### Example Question #1 : Graphical Behavior Of Asymptotes

Where are the vertical asymptotes on the graph of the function

Explanation:

To solve this, find where the function cannot exist. Here, the function cannot exist if the denominator is zero. This happens at x=2 and x=-2. Graph the function on a graphing calculator or by hand to see that the function never crosses these vertical lines. It only gets infinitely close. This is characteristic of vertical asymptotes.

### Example Question #1 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Find all vertical asymptotes and horizontal asymptotes of the function,

Horizontal Aysmptotes

None

Vertical Asymptotes

Horizontal Aysmptotes

Vertical Asymptotes

The horizontal asymptote is,

The vertical asymptotes are,

Horizontal Aysmptotes

Vertical Asymptotes

The horizontal asymptote is,

The vertical asymptotes are,

The horizontal asymptote is,

The vertical asymptotes are,

Explanation:

1) To find the horizontal asymptotes, find the limit of the function as

Therefore, the function  has a horizontal asymptote

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2) Vertical asympototes will occur at points where the function blows up, . For rational functions this behavior occurs when the denominator approaches zero.

Factor the denominator and set to zero,

So the graph of  has two vertical asymptotes, one at  and the other at .  They have been drawn into the graph of   below. The blue curves represent

### Example Question #2 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Undefined.

Explanation:

For this infinity limit, we need to consider the leading terms of both the numerator and the denominator.  In our problem, the leading term of the numerator is larger than the leading term of the denominator.  Therefore, it will be growing at a faster rate.

Now, simply input the limit value, and interpret the results.

### Example Question #3 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Undefined.

Explanation:

For infinity limits, we need only consider the leading term in both the numerator and the denominator.  Here, we have the case that the exponents are equal in the leading terms.  Therefore, the limit at infinity is simply the ratio of the coefficients of the leading terms.

### Example Question #4 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Undefined.

Explanation:

Infinity limits can be found by only considering the leading term in both the numerator and the denominator. In this problem, the numerator has a higher exponent than the denominator.  Therefore, it will keep increasing and increasing at a much faster rate.  These limits always tend to infinity.

### Example Question #5 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Undefined.

Explanation:

For infinity limits, we only consider the leading term in both the numerator and the denominator.  Then, we need to consider the exponents of the leading terms.  Here, the denominator has a higher degree than the numerator.  Therefore, we have a bottom heavy fraction.  Even though we are evaluating the limit at negative infinity, this will still tend to zero because the denominator is growing at a faster rate.  You can convince yourself of this by plugging in larger and larger negative values.  You will just get a longer and smaller decimal.