QuestionFind the limit as $x$ approaches 0 for the expression $x^{4}-\frac{4^{x}}{8000}$ .

Studdy Solution

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.
$\lim{x \rightarrow0}\left(x^{4}-\frac{4^{x}}{8,000}\right) =0 - \frac{1}{8,000}$

STEP 6

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 $-\frac{1}{8,000}$ .

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QuestionFind the limit as $x$ approaches 0 for the expression $x^{4}-\frac{4^{x}}{8000}$ .

Studdy Solution

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.
$\lim{x \rightarrow0}\left(x^{4}-\frac{4^{x}}{8,000}\right) =0 - \frac{1}{8,000}$

STEP 6

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 $-\frac{1}{8,000}$ .

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QuestionFind the limit as xxx approaches 0 for the expression x4−4x8000x^{4}-\frac{4^{x}}{8000}x4−80004x​.

Studdy Solution

STEP 1

Assumptions1. We are asked to find the limit as xxx approaches 000 of the expression x4−4x8,000x^{4}-\frac{4^{x}}{8,000}x4−8,0004x​. . We will use the properties of limits to solve this problem.

STEP 2

STEP 3

The limit of xx^{}x as xxx approaches 000 is simply 00^{}0, which is 000.lim⁡x→0x=0=0\lim{x \rightarrow0}x^{} =0^{} =0limx→0x=0=0

STEP 4

STEP 5

Substitute the limits of the individual terms back into the equation.lim⁡x→0(x4−4x8,000)=0−18,000\lim{x \rightarrow0}\left(x^{4}-\frac{4^{x}}{8,000}\right) =0 - \frac{1}{8,000}limx→0(x4−8,0004x​)=0−8,0001​

STEP 6

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QuestionFind the limit as $x$ approaches 0 for the expression $x^{4}-\frac{4^{x}}{8000}$ .

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.

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

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