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Chapter 3: Problem 53

Evaluate the following limits or state that they do not exist. $$\lim _{x \rightarrow 0} \frac{3 \sec ^{5} x}{x^{2}+4}$$

Short Answer

Expert verified
Question: Find the limit of the following function as x approaches 0: $$\lim _{x \rightarrow 0} \frac{3 \sec ^{5} x}{x^{2}+4}$$ Answer: The limit of the function as x approaches 0 is: $$\lim _{x \rightarrow 0} \frac{3 \sec ^{5} x}{x^{2}+4} = \frac{3}{4}$$

Step by step solution

01

Rewrite the function using the trigonometric identity

Recall that $$\sec x = \frac{1}{\cos x}$$. Then, we can rewrite the given function as $$\frac{3}{(\cos x)^5 (x^2 + 4)}$$.
02

Direct substitution

To find the limit as x approaches 0, we can try to substitute the value of x as 0 directly into the function: $$\lim_{x \rightarrow 0} \frac{3}{(\cos x)^5 (x^2 +4)} = \frac{3}{(\cos 0)^5 (0^2 + 4)}$$
03

Evaluate the limit

Now we can evaluate the cosine value and the remaining expression. $$\lim_{x \rightarrow 0} \frac{3}{(\cos x)^5 (x^2 +4)} = \frac{3}{(1)^5 (0 + 4)} = \frac{3}{4}$$ So, the limit of the given function as x approaches 0 is: $$\lim _{x \rightarrow 0} \frac{3 \sec ^{5} x}{x^{2}+4} = \frac{3}{4}$$

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