Find the limit of $\frac{\log_{3}\left(x^2\right)}{e^x}$ as $x$ approaches $\infty $

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Change the logarithm to base $e$ applying the change of base formula for logarithms: $\log_b(a)=\frac{\log_x(a)}{\log_x(b)}$

$\lim_{x\to\infty }\left(\frac{\frac{\ln\left(x^2\right)}{\ln\left(3\right)}}{e^x}\right)$

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$\lim_{x\to\infty }\left(\frac{\frac{\ln\left(x^2\right)}{\ln\left(3\right)}}{e^x}\right)$

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Learn how to solve limits to infinity problems step by step online. Find the limit of log3(x^2)/(e^x) as x approaches infinity. Change the logarithm to base e applying the change of base formula for logarithms: \log_b(a)=\frac{\log_x(a)}{\log_x(b)}. Divide fractions \frac{\frac{\ln\left(x^2\right)}{\ln\left(3\right)}}{e^x} with Keep, Change, Flip: \frac{a}{b}\div c=\frac{a}{b}\div\frac{c}{1}=\frac{a}{b}\times\frac{1}{c}=\frac{a}{b\cdot c}. If we directly evaluate the limit \lim_{x\to\infty }\left(\frac{\ln\left(x^2\right)}{\ln\left(3\right)e^x}\right) as x tends to \infty , we can see that it gives us an indeterminate form. We can solve this limit by applying L'Hôpital's rule, which consists of calculating the derivative of both the numerator and the denominator separately.

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Function Plot

Plotting: $\frac{\log_{3}\left(x^2\right)}{e^x}$

Main Topic: Limits to Infinity

The limit of a function f(x) when x tends to infinity is the value that the function takes as the value of x grows indefinitely.

Used Formulas

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