Limits and Asymptotes

Section 1.7 Limits and Asymptotes

Subsection 1.7.1 Recommended Tutorials

Before starting on these exercises, you should familiarize yourself with the material covered in the following tutorial:

Limits

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Subsection 1.7.2 Introduction

It is incorrect to assume that a vertical asymptote is always found whenever the denominator of a rational function is equal to zero. Instead, we say that \(f(x)\) has a vertical asymptote at \(x=a\) whenever

\begin{equation*} \dlim{x}{a^+} f(x) = \infty \text{ or } \dlim{x}{a^-} f(x) = \infty\text{.} \end{equation*}

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In either case, the equation of the vertical asymptote is \(x=a\text{.}\)

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Similarly, a horizontal asymptote of \(f(x)\) is also defined in terms of limits. A function \(f(x)\) has a horizontal asymptote \(y = L\) if

\begin{equation*} \dlim{x}{\infty} f(x) = L \text{ or } \dlim{x}{-\infty} f(x) = L\text{.} \end{equation*}

In this case, the equation of the horizontal asymptote is \(y=L\text{.}\)

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Aside

You will need to use both of these definitions while answering the following exercises.

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Exercises 1.7.3 Exercises

1.

Assign the function \(f(x)=\dfrac{x-1}{x^2-x-2}\) using the assignment operator :=.

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Aside

(a)

Plot the function and visually try to determine the \(x\)-values of any vertical asymptotes.

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(b)

Evaluate the left- and right-hand limits on either side of each value from the previous step using limit() commands. If Maple has outputted \(\infty\) or \(-\infty\) as the value of these limits, then you have correctly determined the location of a vertical asymptote.

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2.

Assign the function \(g(x)=\dfrac{x+2}{x^2-x-6}\text{.}\)

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(a)

Factor the denominator of \(g(x)\) to determine any values where the function has a discontinuity.

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Hint.

Maple provides a denom() command as a handy way to get the denominator of an expression. By nesting multiple commands, you should not need to retype any part of the function:

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> factor(denom(g(x)));

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(b)

Remember that just because a rational expression has a discontinuity at a certain value, it does not mean that there is a vertical asymptote at that value! Plot \(g(x)\) so that both discontinuities are visible. How many asymptotes does \(g(x)\) have?

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3.

Assign the function \(h(t)=\dfrac{\sin(t)}{t}\text{.}\)

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(a)

Plot a graph of \(h(t)\text{.}\) Adjust the bounds for \(x\) and for \(y\) so that the graph shows a reasonable amount of detail and gives you and idea of any horizontal asymptotes.

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(b)

Use the limit() command to find the horizontal asymptote(s) of \(h(t)\text{.}\)

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Hint.

An example of finding horizontal asymptotes is provided in Example 11.2.

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4.

Assign the function \(Q(x)=\dfrac{\sqrt{2x^2+1}}{3x+5}\text{.}\)

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Aside

(a)

Plot a graph of \(Q(x)\text{.}\) Be sure to specify appropriate intervals for the \(x\)-axis and \(y\)-axis so that you can observe the shape of the function as well as whether it has any horizontal asymptotes.

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(b)

Use the limit() command to find the horizontal asymptote(s) of \(Q(x)\text{.}\)

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