Math Snap

STEP 1

Assumptions1. We are asked to find the limit as $x$ approaches $0$ of the expression $x^{4}-\frac{4^{x}}{8,000}$.
. We will use the properties of limits to solve this problem.

STEP 2

We will find the limit of each term separately, as the limit of a difference is the difference of the limits.
$$\lim{x \rightarrow0}\left(x^{4}-\frac{4^{x}}{8,000}\right) = \lim{x \rightarrow0}x^{4} - \lim{x \rightarrow0}\frac{4^{x}}{8,000}$$

STEP 3

The limit of $x^{}$ as $x$ approaches $0$ is simply $0^{}$, which is $0$.
$$\lim{x \rightarrow0}x^{} =0^{} =0$$

STEP 4

The limit of $\frac{4^{x}}{8,000}$ as $x$ approaches $0$ is $\frac{4^{0}}{8,000}$, which is $\frac{1}{8,000}$.
$$\lim{x \rightarrow0}\frac{4^{x}}{8,000} = \frac{4^{0}}{8,000} = \frac{1}{8,000}$$

STEP 5

Substitute the limits of the individual terms back into the equation.
$$\lim{x \rightarrow0}\left(x^{4}-\frac{4^{x}}{8,000}\right) =0 - \frac{1}{8,000}$$

implify the expression to find the final value of the limit.
$$\lim{x \rightarrow0}\left(x^{4}-\frac{4^{x}}{8,000}\right) = -\frac{1}{8,000}$$The value of the limit as $x$ approaches $0$ of the expression $x^{4}-\frac{4^{x}}{8,000}$ is $-\frac{1}{8,000}$.