limit(f(x)-f(x-1), x=oo)
︠e6221009-8faa-4cad-8785-dda1a0620e7ai︠
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As expected, the limit of the function's slope as x approaches infinity is 1. A hunch would tell you that the function's slope as x approaches negative infinity is most likely 1 as well, and changing the x=oo to x=-oo verifies that result.
Now that you've done things the hard way, though, I'll tell you a shortcut to find the slope of slant asymptotes for rational functions. For a generalized rational function like this one:
$\begin{align}f(x)=\frac{ax^{n+1}+\ldots}{bx^n+\ldots}\end{align}$
If n is the highest power of the denominator, n+1 is the highest power of the numerator, and a and b are constants, the function will have a slant asymptote with a slope equal to a/b. You will find that slant asymptotes only pop up when the numerator of a function is of one higher power than the denominator of a rational function. Where numerical analysis can still come into play, though, in a case where you can't simplify a function to fit this general form.
Determine the slope, if possible, of each function as it approaches positive and negative infinity.
$\begin{align}\mbox{1) }h(x)=\frac{5x^3-2x}{x^2+1}\end{align}$
$\begin{align}\mbox{2) }g(x)=\frac{1-\sin^2{x}+x}{\cos^2{x}}\end{align}$
$\begin{align}\mbox{3) Hint: simplify first. }h(x)=\frac{1}{x}+x+\frac{x^2-1}{x}+\frac{x}{x^2-1}\end{align}$
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As expected, the limit of the function's slope as x approaches infinity is 1. A hunch would tell you that the function's slope as x approaches negative infinity is most likely 1 as well, and changing the x=oo to x=-oo verifies that result.

Now that you've done things the hard way, though, I'll tell you a shortcut to find the slope of slant asymptotes for rational functions. For a generalized rational function like this one:

If n is the highest power of the denominator, n+1 is the highest power of the numerator, and a and b are constants, the function will have a slant asymptote with a slope equal to a/b. You will find that slant asymptotes only pop up when the numerator of a function is of one higher power than the denominator of a rational function. Where numerical analysis can still come into play, though, in a case where you can't simplify a function to fit this general form.

## Practice Problems

Determine the slope, if possible, of each function as it approaches positive and negative infinity.

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