evaluate the limit using lhospitals rule if necessary\n\\(\\lim_{x\\to\\infty}(16x)^{\\frac{\\ln 10 +…

Explanation:

Step1: Take natural - logarithm of the function

Let $y=(16x)^{\frac{\ln 10 + 1}{\ln(3x)+1}}$. Then $\ln y=\frac{\ln 10 + 1}{\ln(3x)+1}\ln(16x)$. As $x\rightarrow\infty$, we have an indeterminate form of type $\frac{\infty}{\infty}$. $$\ln y=\frac{(\ln 10 + 1)\ln(16x)}{\ln(3x)+1}$$

Step2: Apply L'Hopital's rule

Differentiate the numerator and denominator. The derivative of $(\ln 10 + 1)\ln(16x)$ with respect to $x$ is $\frac{\ln 10 + 1}{x}$, and the derivative of $\ln(3x)+1$ with respect to $x$ is $\frac{1}{x}$. $$\lim_{x\rightarrow\infty}\ln y=\lim_{x\rightarrow\infty}\frac{\frac{\ln 10 + 1}{x}}{\frac{1}{x}}$$

Step3: Simplify the limit of $\ln y$

Cancel out the $x$ terms in the fraction. $$\lim_{x\rightarrow\infty}\ln y=\ln 10 + 1$$

Step4: Find the limit of $y$

Since $\lim_{x\rightarrow\infty}\ln y=\ln 10 + 1$, then $y = e^{\ln 10+1}$. Using the property $a^{m + n}=a^{m}\cdot a^{n}$, we have $y = 10e$.

Answer:

$10e$