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Chapter 3: Problem 12

Use Theorem 3.11 to evaluate the following limits. $$\lim _{x \rightarrow 0} \frac{\tan 5 x}{x}$$

Short Answer

Expert verified
Based on the explanation above, please answer the following question: What is the value of the limit as x approaches 0 for the given expression: $$\lim_{x \rightarrow 0} \frac{\tan 5x}{x}$$ Answer: The value of the given limit is 5.

Step by step solution

01

Rewrite the given expression using properties of limits

The given limit expression is: $$\lim_{x \rightarrow 0} \frac{\tan 5x}{x}$$ In order to simplify this expression, we will rewrite it as the product of the limit: $$\lim_{x \rightarrow 0} \frac{\tan 5x}{5x} \cdot 5$$
02

Use L'H么pital's Rule

To apply L'H么pital's Rule, we need to find the derivatives of the numerator and the denominator with respect to x. So, let's differentiate both the numerator and the denominator: $$\frac{d}{dx}(\tan 5x) = \sec^2(5x) \cdot 5 = 5 \sec^2(5x)$$ $$\frac{d}{dx}(5x) = 5$$ Now, let's apply L'H么pital's Rule to find the limit: $$\lim_{x \rightarrow 0} \frac{\tan 5x}{5x} = \lim_{x \rightarrow 0} \frac{5 \sec^2(5x)}{5}$$
03

Simplify the expression

Simplify the expression by dividing 5 in both numerator and denominator: $$\lim_{x \rightarrow 0} \frac{5 \sec^2(5x)}{5} = \lim_{x \rightarrow 0} \sec^2(5x)$$
04

Evaluate the limit

Using the definition of secant and knowing that the cosine of 0 is 1, we can write: $$\lim_{x \rightarrow 0} \sec^2(5x) = \lim_{x \rightarrow 0} \frac{1}{\cos^2(5x)}$$ As \(x\) approaches 0, \(5x\) also approaches 0, thus: $$\lim_{x \rightarrow 0} \frac{1}{\cos^2(5x)} = \frac{1}{\cos^2(0)} = \frac{1}{1^2} = 1$$
05

Multiply by 5 to obtain the final result

Recall from Step 1 that we multiplied the limit by 5 to rewrite the expression. Now, we will multiply our result by 5 to obtain the final answer: $$\lim_{x \rightarrow 0} \frac{\tan 5x}{x} = 1 \cdot 5 = 5$$ Therefore, the given limit is 5.

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