QuestionFind the limit: $\lim _{x \rightarrow 0} \frac{\frac{1}{5+x}-\frac{1}{5}}{x}$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to find the limit as x approaches0 of the function $\frac{\frac{1}{5+x}-\frac{1}{5}}{x}$

STEP 2

We can see that if we directly substitute $x=0$ into the function, we get a form of $\frac{0}{0}$ , which is indeterminate. Therefore, we need to simplify the function first.

STEP 3

To simplify the function, we can start by finding a common denominator for the fractions in the numerator.
$\frac{\frac{1}{5+x}-\frac{1}{5}}{x} = \frac{5 - (5+x)}{x(5+x)5}$

STEP 4

implify the numerator.
$\frac{ - (+x)}{x(+x)} = \frac{-x}{x(+x)}$

STEP 5

Now, we can cancel out the $x$ in the numerator and the denominator.
$\frac{-x}{x(5+x)5} = \frac{-1}{(5+x)5}$

STEP 6

Now, we can substitute $x=0$ into the simplified function.
$\lim{x \rightarrow0} \frac{-1}{(5+x)5} = \frac{-1}{(5+0)5}$

STEP 7

Calculate the limit.
$\frac{-1}{(5+0)5} = \frac{-1}{25}$ The limit as x approaches0 of the function $\frac{\frac{1}{5+x}-\frac{1}{5}}{x}$ is $\frac{-1}{25}$ .

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QuestionFind the limit: $\lim _{x \rightarrow 0} \frac{\frac{1}{5+x}-\frac{1}{5}}{x}$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to find the limit as x approaches0 of the function $\frac{\frac{1}{5+x}-\frac{1}{5}}{x}$

STEP 2

We can see that if we directly substitute $x=0$ into the function, we get a form of $\frac{0}{0}$ , which is indeterminate. Therefore, we need to simplify the function first.

STEP 3

To simplify the function, we can start by finding a common denominator for the fractions in the numerator.
$\frac{\frac{1}{5+x}-\frac{1}{5}}{x} = \frac{5 - (5+x)}{x(5+x)5}$

STEP 4

implify the numerator.
$\frac{ - (+x)}{x(+x)} = \frac{-x}{x(+x)}$

STEP 5

Now, we can cancel out the $x$ in the numerator and the denominator.
$\frac{-x}{x(5+x)5} = \frac{-1}{(5+x)5}$

STEP 6

Now, we can substitute $x=0$ into the simplified function.
$\lim{x \rightarrow0} \frac{-1}{(5+x)5} = \frac{-1}{(5+0)5}$

STEP 7

Calculate the limit.
$\frac{-1}{(5+0)5} = \frac{-1}{25}$ The limit as x approaches0 of the function $\frac{\frac{1}{5+x}-\frac{1}{5}}{x}$ is $\frac{-1}{25}$ .

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QuestionFind the limit: lim⁡x→015+x−15x\lim _{x \rightarrow 0} \frac{\frac{1}{5+x}-\frac{1}{5}}{x}limx→0​x5+x1​−51​​.

Studdy Solution

STEP 1

Assumptions1. We are asked to find the limit as x approaches0 of the function 15+x−15x\frac{\frac{1}{5+x}-\frac{1}{5}}{x}x5+x1​−51​​

STEP 2

We can see that if we directly substitute x=0x=0x=0 into the function, we get a form of 00\frac{0}{0}00​, which is indeterminate. Therefore, we need to simplify the function first.

STEP 3

To simplify the function, we can start by finding a common denominator for the fractions in the numerator.15+x−15x=5−(5+x)x(5+x)5\frac{\frac{1}{5+x}-\frac{1}{5}}{x} = \frac{5 - (5+x)}{x(5+x)5}x5+x1​−51​​=x(5+x)55−(5+x)​

STEP 4

implify the numerator.−(+x)x(+x)=−xx(+x)\frac{ - (+x)}{x(+x)} = \frac{-x}{x(+x)}x(+x)−(+x)​=x(+x)−x​

STEP 5

Now, we can cancel out the xxx in the numerator and the denominator.−xx(5+x)5=−1(5+x)5\frac{-x}{x(5+x)5} = \frac{-1}{(5+x)5}x(5+x)5−x​=(5+x)5−1​

STEP 6

Now, we can substitute x=0x=0x=0 into the simplified function.lim⁡x→0−1(5+x)5=−1(5+0)5\lim{x \rightarrow0} \frac{-1}{(5+x)5} = \frac{-1}{(5+0)5}limx→0(5+x)5−1​=(5+0)5−1​

STEP 7

Calculate the limit.−1(5+0)5=−125\frac{-1}{(5+0)5} = \frac{-1}{25}(5+0)5−1​=25−1​The limit as x approaches0 of the function 15+x−15x\frac{\frac{1}{5+x}-\frac{1}{5}}{x}x5+x1​−51​​ is −125\frac{-1}{25}25−1​.

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QuestionFind the limit: $\lim _{x \rightarrow 0} \frac{\frac{1}{5+x}-\frac{1}{5}}{x}$ .

Assumptions1. We are asked to find the limit as x approaches0 of the function $\frac{\frac{1}{5+x}-\frac{1}{5}}{x}$

We can see that if we directly substitute $x=0$ into the function, we get a form of $\frac{0}{0}$ , which is indeterminate. Therefore, we need to simplify the function first.

To simplify the function, we can start by finding a common denominator for the fractions in the numerator.
$\frac{\frac{1}{5+x}-\frac{1}{5}}{x} = \frac{5 - (5+x)}{x(5+x)5}$

implify the numerator.
$\frac{ - (+x)}{x(+x)} = \frac{-x}{x(+x)}$

Now, we can cancel out the $x$ in the numerator and the denominator.
$\frac{-x}{x(5+x)5} = \frac{-1}{(5+x)5}$

Now, we can substitute $x=0$ into the simplified function.
$\lim{x \rightarrow0} \frac{-1}{(5+x)5} = \frac{-1}{(5+0)5}$

Calculate the limit.
$\frac{-1}{(5+0)5} = \frac{-1}{25}$ The limit as x approaches0 of the function $\frac{\frac{1}{5+x}-\frac{1}{5}}{x}$ is $\frac{-1}{25}$ .