To solve the limit [tex]\(\lim_{x \rightarrow 0} 5 x^{\sqrt{9x}}\)[/tex], we need to carefully analyze the behavior of the function [tex]\(5 x^{\sqrt{9x}}\)[/tex] as [tex]\(x\)[/tex] approaches 0.
1. Expression Analysis:
The function we are interested in is [tex]\(5 x^{\sqrt{9x}}\)[/tex]. This is an exponential function with a base [tex]\(x\)[/tex] and an exponent that involves [tex]\(x\)[/tex].
2. Exponent Simplification:
Let's focus on the exponent [tex]\(\sqrt{9x}\)[/tex]. Simplifying, we get,
[tex]\[
\sqrt{9x} = (\sqrt{9} \cdot \sqrt{x}) = 3\sqrt{x}
\][/tex]
Thus, the expression can be rewritten as:
[tex]\[
5 x^{3\sqrt{x}}
\][/tex]
3. Taking the Natural Logarithm:
To handle the complex exponentiation, we can take the natural logarithm of the function to simplify the exponent. Set [tex]\(y = x^{3\sqrt{x}}\)[/tex]. Then,
[tex]\[
\ln y = 3\sqrt{x} \cdot \ln x
\][/tex]
4. Limit of the Exponential Part:
We now focus on finding the limit of [tex]\(\ln y\)[/tex] as [tex]\(x\)[/tex] approaches 0. Consider the term [tex]\(3\sqrt{x} \cdot \ln x\)[/tex]:
[tex]\[
\lim_{x \to 0} 3\sqrt{x} \ln x
\][/tex]
5. Substitution for Simplification:
Let's perform a substitution to simplify the analysis. Set [tex]\(u = \sqrt{x}\)[/tex]. Hence, when [tex]\(x\)[/tex] approaches 0, [tex]\(u\)[/tex] also approaches 0. The expression [tex]\(3\sqrt{x} \ln x\)[/tex] becomes:
[tex]\[
3u \ln (u^2) = 3u (2 \ln u) = 6u \ln u
\][/tex]
6. Evaluating the Limit:
We need to evaluate:
[tex]\[
\lim_{u \to 0} 6u \ln u
\][/tex]
Notice that as [tex]\(u\)[/tex] approaches 0, [tex]\(\ln u\)[/tex] approaches [tex]\(-\infty\)[/tex], and [tex]\(u\)[/tex] approaches 0; thus, [tex]\(6u \ln u\)[/tex] approaches 0.
7. Applying the Limit:
Since [tex]\(\ln y = 6u \ln u\)[/tex], it follows that:
[tex]\[
\lim_{x \to 0} \ln x^{3\sqrt{x}} = \lim_{u \to 0} 6u \ln u = 0
\][/tex]
8. Exponentiation to Reverse the Logarithm:
If [tex]\(\ln L = 0\)[/tex], then [tex]\(L = e^0 = 1\)[/tex]. Consequently:
[tex]\[
\lim_{x \to 0} x^{3\sqrt{x}} = 1
\][/tex]
9. Include the Constant Factor:
Therefore, taking into account the original multiplier:
[tex]\[
\lim_{x \rightarrow 0} 5 x^{3 \sqrt{x}} = 5 \times 1 = 5
\][/tex]
The limit of the function as [tex]\(x\)[/tex] approaches 0 is:
[tex]\[
\boxed{5}
\][/tex]
