To solve the limit [tex]\(\lim_{x \rightarrow 0} \frac{\tan(x^g)}{x}\)[/tex], we can follow a detailed step-by-step process:
1. Understand the function: We are given the limit of [tex]\(\frac{\tan(x^g)}{x}\)[/tex] as [tex]\(x\)[/tex] approaches 0. Here, [tex]\(g\)[/tex] is a constant exponent.
2. Rewrite the expression: We recognize that [tex]\(x^g\)[/tex] is another function of [tex]\(x\)[/tex] and let [tex]\(u = x^g\)[/tex]. As [tex]\(x\)[/tex] approaches 0, [tex]\(u\)[/tex] will also approach 0.
3. Substitute [tex]\(u\)[/tex] in the limit: The original limit can now be rewritten in terms of [tex]\(u\)[/tex]:
[tex]\[
\frac{\tan(u)}{x}
\][/tex]
where [tex]\(u = x^g\)[/tex]. Hence,
[tex]\[
x = u^{1/g}
\][/tex]
Substituting this back into the limit, we get:
[tex]\[
\frac{\tan(u)}{u^{1/g}}
\][/tex]
4. Analyze the limit behavior: To understand the behavior of this expression as [tex]\(x \rightarrow 0\)[/tex] (or equivalently [tex]\(u \rightarrow 0\)[/tex]), recall that [tex]\(\tan(u)\)[/tex] is approximately [tex]\(u\)[/tex] for small values of [tex]\(u\)[/tex]. Thus,
[tex]\[
\tan(u) \approx u \quad \text{as} \quad u \rightarrow 0
\][/tex]
5. Simplify the expression: Using the approximation [tex]\(\tan(u) \approx u\)[/tex]:
[tex]\[
\frac{\tan(u)}{u^{1/g}} \approx \frac{u}{u^{1/g}} = u^{1 - 1/g}
\][/tex]
6. Evaluate the limit: As [tex]\(u \rightarrow 0\)[/tex],
[tex]\[
u^{1 - 1/g} \rightarrow 0 \quad \text{if} \quad 1 - \frac{1}{g} > 0
\][/tex]
This simplifies to:
[tex]\[
1 - \frac{1}{g} > 0 \quad \Rightarrow \quad g > 1
\][/tex]
Conversely, if [tex]\(g = 1\)[/tex],
[tex]\[
u^{1 - 1} = u^0 = 1
\][/tex]
And if [tex]\(g < 1\)[/tex], as [tex]\(u \rightarrow 0\)[/tex],
[tex]\[
u^{1 - 1/g} \rightarrow \infty
\][/tex]
We are particularly interested in evaluating the limit when the exponent [tex]\(g\)[/tex] is such that the terms balance out correctly. For [tex]\(g = 1\)[/tex],
[tex]\[
\frac{\tan(x^g)}{x} = \frac{\tan(x)}{x}
\][/tex]
7. Final answer: We know that [tex]\(\lim_{x \rightarrow 0} \frac{\tan x}{x} = 1\)[/tex]. Therefore,
[tex]\[
\lim_{x \rightarrow 0} \frac{\tan(x^g)}{x} = 1 \quad \text{when} \quad g = 1
\][/tex]
Hence, the answer for the given question is:
[tex]\[
1
\][/tex]
