Single Variable Calculus, Chapter 4, 4.4, Section 4.4, Problem 58

Sketch the curve $y = x^n$ ($n$ an integer) for the following five cases:

i) $n = 0$







ii) $n > 0, n $ odd







iii) $n > 0, n$ even







iv) $n < 0, n$ odd







v) $n < 0, n$ even







Then use these sketches to find the following limits:

a) $\lim_{x \to 0^+} x^n$

Referring to the graphs,

$\lim_{x \to 0^+} x^n = \left\{
\begin{array}{cc}
1 & \text{if } n = 0 \\
0 & \text{if } n > 0, n \text{ is odd} \\
0 & \text{if } n > 0, n \text{ is even} \\
\infty & \text{if } n < 0, n \text{ is odd} \\
\infty & \text{if } n < 0, n \text{ is even}
\end{array}
\right.$

b) $\lim_{x \to 0^-} x^n$

Referring to the graphs,

$\lim_{x \to 0^-} x^n = \left\{
\begin{array}{cc}
1 & \text{if } n = 0 \\
0 & \text{if } n > 0, n \text{ is odd} \\
0 & \text{if } n > 0, n \text{ is even} \\
- \infty & \text{if } n < 0, n \text{ is odd} \\
\infty & \text{if } n < 0, n \text{ is even}
\end{array}
\right.$

c) $\lim_{x \to \infty} x^n$

Referring to the graphs,

$\lim_{x \to \infty} x^n = \left\{
\begin{array}{cc}
1 & \text{if } n = 0 \\
\infty & \text{if } n > 0, n \text{ is odd} \\
\infty & \text{if } n > 0, n \text{ is even} \\
0 & \text{if } n < 0, n \text{ is odd} \\
0 & \text{if } n < 0, n \text{ is even}
\end{array}
\right.$

d) $\lim_{x \to -\infty} x^n$

Referring to the graphs,

$\lim_{x \to - \infty} x^n = \left\{
\begin{array}{cc}
1 & \text{if } n = 0 \\
- \infty & \text{if } n > 0, n \text{ is odd} \\
\infty & \text{if } n > 0, n \text{ is even} \\
0 & \text{if } n < 0, n \text{ is odd} \\
0 & \text{if } n < 0, n \text{ is odd}
\end{array}
\right.$