Chapter 12.4, Problem 69E

To determine

Explain why $y_1$ not appear to the left of the y-axis?

Answer to Problem 69E

The $\sqrt{x}$ is undefined for $x<0$ , the limit $\underset{x\to -\infty }{\lim }\frac{1}{\sqrt{x}}$ is undefined.

Explanation of Solution

Given information:

Use a graphing utility to graph the function

  $y_1=\frac{1}{\sqrt{x}}$

  $and$

  $y_2=\frac{1}{\sqrt[3]{x}}$

in the same viewing window. Why does $y_1$ not appear to the left of the y-axis? How does this relate to the statement about the infinite limit $\underset{x\to \infty }{\lim }\frac{1}{x^r}$ ?

Calculation:

Consider the following funnction

  $y_1=\frac{1}{\sqrt{x}}$

  $and$

  $y_2=\frac{1}{\sqrt[3]{x}}$

To draw the graph of the above funcion, use $TI-83$ plus graphing calculator on the same window. Follow the step by step procedure:

Press $\boxed{Y=}$ ,write the function and press $\boxed{ENTER}$ . To write cube roots, press the key $\boxed{MATH}$ And choose the option $4:\sqrt[3]{}($ under MATH.

  

Press $\boxed{WINDOW}$ and adjust the $x$ and $y$ ranges. Also adjust the step sizes.

  

To draw the graph, press the $\boxed{GRAPH}$ .

  

We see that $y_1$ does not appear on the left of the $y-axis$ . The reason is that $\sqrt{x}$ is undefined for $x<0$ .

If $x^r$ is defined when $x<0$ , then

  $\underset{x\to -\infty }{\lim }\frac{1}{x^r}=0$

Hence, the $\sqrt{x}$ is undefined for $x<0$ , the limit $\underset{x\to -\infty }{\lim }\frac{1}{\sqrt{x}}$ is undefined.