Chapter 11.2, Problem 76E

To determine

To state: which Limit is evaluated by direct substitution and find the limit

Answer to Problem 76E

Limit (a) can be evaluated by direct substitution, the $\underset{x\to 0}{\lim }\frac{x}{\cos \text{}\text{}x}=0$ and $\underset{x\to 0}{\lim }\frac{1-\cos \text{}\text{}x}{\cos \text{}\text{}x}=0$

Explanation of Solution

Given:

  $\underset{x\to 0}{\lim }\frac{x}{\cos \text{}x}$ and $\underset{x\to 0}{\lim }\frac{1-\cos \text{}\text{}x}{\cos \text{}\text{}x}$

Calculation:

Limit (a) can be evaluated by using direct substitution, since it is a rational function defined at $x=0$ . Therefore

  $\underset{x\to 0}{\lim }\frac{x}{\cos \text{}x}=\frac{0}{\cos 0}=\frac{0}{1}=0$

Limit (b) cannot be evaluated by direct substitution, since the function is undefined at $x=0$ (causes a divide by zero), so approximate this limit by using a table:

$x$	$f\left(x\right)$
$-0.1$$-0.01$$-0.001$$0$$0.001$$0.01$$0.1$	$-0.0499583$$-0.00499996$$-0.0005$ Indeterminate $0.0005$$0.00499996$$0.0499583$

From the table, see that the function approaches the value $1$ as $x\to 0$ from both the positive and negative directions, so

  $\underset{x\to 0}{\lim }\frac{1-\cos \text{}\text{}x}{\cos \text{}\text{}x}=0$

Limit (a) can be evaluated by direct substitution.

  $\underset{x\to 0}{\lim }\frac{x}{\cos \text{}\text{}x}=0$

  $\underset{x\to 0}{\lim }\frac{1-\cos \text{}\text{}x}{\cos \text{}\text{}x}=0$

Conclusion:

Therefore,Limit (a) can be evaluated by direct substitution, $\underset{x\to 0}{\lim }\frac{x}{\cos \text{}\text{}x}=0$ and $\underset{x\to 0}{\lim }\frac{1-\cos \text{}\text{}x}{\cos \text{}\text{}x}=0$.