Evaluate $\displaystyle \large \lim_{x\,\to\,\infty}{\normalsize \bigg(1+\dfrac{4}{x}\bigg)^{\displaystyle 3x}}$

The limit of $1$ plus the quotient of $4$ by $x$ whole raised to the power of $3x$ should be evaluated as the value of $x$ is closer to infinity in this problem.

$\displaystyle \large \lim_{x\,\to\,\infty}{\normalsize \bigg(1+\dfrac{4}{x}\bigg)^{\displaystyle 3x}}$

Let’s try to find the limit of the function in exponential notation by the direct substitution.

$=\,\,\,$ $\bigg(1+\dfrac{4}{\infty}\bigg)^{\displaystyle 3(\infty)}$

$=\,\,\,$ $(1+0)^{\displaystyle \infty}$

$=\,\,\,$ $(1)^{\displaystyle \infty}$

$=\,\,\,$ $1^{\displaystyle \infty}$

It is evaluated that the limit of one plus four by $x$ whole raised to the power of three times $x$ as the value of $x$ approaches to infinity is equal to one raised to the power of infinity. It is an indeterminate form.

Hence, we must explore for alternative methods to find the limit of the given function as the value of $x$ tends to infinity.

Fundamental

Learn how to find the limit of $1$ plus $4$ by $x$ whole power of $3x$ as the value of $x$ approaches to infinity by eliminating the exponential notation with logarithms.

Limit Rules

Learn how to evaluate the limit of $1$ plus $4$ by $x$ whole power of $3x$ as the value of $x$ tends to infinity by the limit rules.