Q.2 - Calculus

Question

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) Three limits of the indeterminate form $\frac{0}{0}$, one that approaches  $\infty$, one that equals $0$, and one that equals $3$.

(b) Three limits of the indeterminate form $\frac{\infty }{\infty }$, one that approaches $\infty$, one that equals $0$, and one that equals $3$.

(c) Three limits of the indeterminate form $0.\infty$, one that approaches $\infty$, one that equals $0$, and one that equals $3$.

Step-by-Step Solution

Answer

(a) The three limits are: $\underset{x\to 0}{lim}\frac{x}{x^2},\underset{x\to 0}{lim}\frac{x^2}{x},\underset{x\to 0}{lim}\frac{x^2+3x}{x}$

(b) The three limits are: $\underset{x\to 0}{lim}\frac{x^2}{x}, \underset{x\to \infty }{lim}\frac{x}{x^2}, \underset{x\to \infty }{lim}\frac{3x}{x+1}$

(c) The three limits are: $\underset{x\to 0}{lim}x.\frac{1}{x^2}, \underset{x\to \infty }{lim}{2}^{-x}.x,\underset{x\to 0}{lim}(x^2+3x).\frac{1}{x}$

Part (a) Step 1. Given Information.

Three limits of the indeterminate form $\frac{\infty }{\infty }$, one that approaches  $\infty$, one that equals $0$, and one that equals $3$.

Part (a) Step 2. Find three limits.

By using L'Hopital's rule,

$$\begin{gathered} \underset{x\to 0}{lim}\frac{x}{x^2}=\underset{x\to 0}{lim}\frac{1}{2x} \\ =\frac{1}{0} \\ =\infty \end{gathered}$$

$$\begin{gathered} \underset{x\to 0}{lim}\frac{x^2}{x}=\underset{x\to 0}{lim}\frac{2x}{1} \\ =0 \\ \underset{x\to 0}{lim}\frac{x^2+3x}{x}=\underset{x\to 0}{lim}\frac{2x+3}{1} \\ =3 \end{gathered}$$

Part (b) Step 2. Find three limit.

$$\begin{gathered} \underset{x\to 0}{lim}\frac{x^2}{x}=\underset{x\to 0}{lim}\frac{2x}{1} \\ =\infty \\ \underset{x\to \infty }{lim}.\frac{x}{x^2}=\underset{x\to \infty }{lim}\frac{x}{x^2} \\ =\underset{x\to 0}{lim}\frac{1}{2x} \\ =0 \\ \underset{x\to \infty }{lim}\frac{3x}{x+1}=\underset{x\to \infty }{lim}\frac{3}{1} \\ =3 \end{gathered}$$

Part (c) Step 1. Find the limits.

$$\begin{gathered} \underset{x\to 0}{lim}x.\frac{1}{x^2}=\underset{x\to 0}{lim}\frac{x}{x^2} \\ =\underset{x\to 0}{lim}\frac{1}{2x} \\ =\infty \\ \underset{x\to \infty }{lim}{2}^{-x}.x=\underset{x\to \infty }{lim}\frac{x}{2^x} \\ =\underset{x\to 0}{lim}\frac{1}{2^x\ln 2} \\ =0 \\ \underset{x\to 0}{lim}(x^2+3x)\frac{1}{x}=\underset{x\to 0}{lim}\frac{x^2+3x}{x} \\ =\underset{x\to 0}{lim}\frac{2x+3}{1} \end{gathered}$$