A vertical asymptote is a place where the function becomes infinite, typically because the formula for the function has a denominator that becomes zero. For example, the reciprocal function $f(x)=1/x$ has a vertical asymptote at $x=0$, and the function $\tan x$ has a vertical asymptote at $x=\pi/2$ (and also at $x=-\pi/2$, $x=3\pi/2$, etc.). Whenever the formula for a function contains a denominator it is worth looking for a vertical asymptote by checking to see if the denominator can ever be zero, and then checking the limit at such points. Note that there is not always a vertical asymptote where the denominator is zero: $f(x)=(\sin x)/x$ has a zero denominator at $x=0$, but since $\ds \lim_{x\to 0}(\sin x)/x=1$ there is no asymptote there.

A horizontal asymptote is a horizontal line to which $f(x)$ gets closer and closer as $x$ approaches $\infty$ (or as $x$ approaches $-\infty$). For example, the reciprocal function has the $x$-axis for a horizontal asymptote. Horizontal asymptotes can be identified by computing the limits $\ds \lim_{x \to \infty}f(x)$ and $\ds \lim_{x \to -\infty}f(x)$. Since $\ds \lim_{x \to \infty}1/x=\lim_{x \to -\infty}1/x=0$, the line $y=0$ (that is, the $x$-axis) is a horizontal asymptote in both directions.

Some functions have asymptotes that are neither horizontal nor vertical, but some other line. Such asymptotes are somewhat more difficult to identify and we will ignore them.

If the domain of the function does not extend out to infinity, we should also ask what happens as $x$ approaches the boundary of the domain. For example, the function $\ds y=f(x)=1/\sqrt{r^2-x^2}$ has domain $-r< x< r$, and $y$ becomes infinite as $x$ approaches either $r$ or $-r$. In this case we might also identify this behavior because when $x=\pm r$ the denominator of the function is zero.

If there are any points where the derivative fails to exist (a cusp or corner), then we should take special note of what the function does at such a point.

Finally, it is worthwhile to notice any symmetry. A function $f(x)$ that has the same value for $-x$ as for $x$, i.e., $f(-x)=f(x)$, is called an "even function.'' Its graph is symmetric with respect to the $y$-axis. Some examples of even functions are: $\ds x^n$ when $n$ is an even number, $\cos x$, and $\ds \sin^2x$. On the other hand, a function that satisfies the property $f(-x)=-f(x)$ is called an "odd function.'' Its graph is symmetric with respect to the origin. Some examples of odd functions are: $x^n$ when $n$ is an odd number, $\sin x$, and $\tan x$. Of course, most functions are neither even nor odd, and do not have any particular symmetry.

Exercises 5.5

Sketch the curves. Identify clearly any interesting features, including local maximum and minimum points, inflection points, asymptotes, and intercepts. You can use this Sage worksheet to check your answers. Note that you may need to adjust the interval over which the function is graphed to capture all the details.

Ex 5.5.1 $\ds y=x^5-5x^4+5x^3$

Ex 5.5.2 $\ds y=x^3-3x^2-9x+5$

Ex 5.5.3 $\ds y=(x-1)^2(x+3)^{2/3}$

Ex 5.5.4 $\ds x^2+x^2y^2=a^2y^2$, $a>0$.

Ex 5.5.5 $\ds y=xe^x$

Ex 5.5.6 $\ds y=(e^x+e^{-x})/2$

Ex 5.5.7 $\ds y=e^{-x}\cos x$

Ex 5.5.8 $\ds y=e^x-\sin x$

Ex 5.5.9 $\ds y=e^x/x$

Ex 5.5.10 $\ds y = 4x+\sqrt{1-x}$

Ex 5.5.11 $\ds y = (x+1)/\sqrt{5x^2 + 35}$

Ex 5.5.12 $\ds y= x^5 - x$

Ex 5.5.13 $\ds y = 6x + \sin 3x$

Ex 5.5.14 $\ds y = x+ 1/x$

Ex 5.5.15 $\ds y = x^2+ 1/x$

Ex 5.5.16 $\ds y = (x+5)^{1/4}$

Ex 5.5.17 $\ds y = \tan^2 x$

Ex 5.5.18 $\ds y =\cos^2 x - \sin^2 x$

Ex 5.5.19 $\ds y = \sin^3 x$

Ex 5.5.20 $\ds y=x(x^2+1)$

Ex 5.5.21 $\ds y=x^3+6x^2 + 9x$

Ex 5.5.22 $\ds y=x/(x^2-9)$

Ex 5.5.23 $\ds y=x^2/(x^2+9)$

Ex 5.5.24 $\ds y=2\sqrt{x} - x$

Ex 5.5.25 $\ds y=3\sin(x) - \sin^3(x)$, for $x\in[0,2\pi]$

Ex 5.5.26 $\ds y=(x-1)/(x^2)$

For each of the following five functions, identify any vertical and horizontal asymptotes, and identify intervals on which the function is concave up and increasing; concave up and decreasing; concave down and increasing; concave down and decreasing.

Ex 5.5.27 $f(\theta)=\sec(\theta)$

Ex 5.5.28 $\ds f(x) = 1/(1+x^2)$

Ex 5.5.29 $\ds f(x) = (x-3)/(2x-2)$

Ex 5.5.30 $\ds f(x) = 1/(1-x^2)$

Ex 5.5.31 $\ds f(x) = 1+1/(x^2)$

Ex 5.5.32 Let $\ds f(x) = 1/(x^2-a^2)$, where $a\geq0$. Find any vertical and horizontal asymptotes and the intervals upon which the given function is concave up and increasing; concave up and decreasing; concave down and increasing; concave down and decreasing. Discuss how the value of $a$ affects these features.