QuestionCalculate the limit: $\lim _{x \rightarrow e}(x \ln x - x)$

Studdy Solution

STEP 1

Assumptions1. We are dealing with a limit problem where $x$ approaches $e$ . . The function is $x \ln x - x$ .
3. We are using the natural logarithm, which has a base of $e$ .

STEP 2

We can see that as $x$ approaches $e$ , both $x \ln x$ and $x$ become indeterminate forms. This suggests that we can use L'Hopital's rule to solve the limit. L'Hopital's rule states that if the limit of a function leads to an indeterminate form (0/0 or ∞/∞), then the limit of that function is equal to the limit of the derivative of the numerator divided by the derivative of the denominator.
However, before we can apply L'Hopital's rule, we need to rewrite the function in the form of a fraction.

STEP 3

Rewrite the function $x \ln x - x$ as a fraction.
$\lim{x \rightarrow e}\frac{x \ln x - x}{1}$

STEP 4

Now, we can apply L'Hopital's rule. This means we need to take the derivative of the numerator and the derivative of the denominator.
The derivative of the numerator, $x \ln x - x$ , can be found using the product rule and the chain rule. The derivative of the denominator, $1$ , is $0$ .

STEP 5

Find the derivative of the numerator, $x \ln x - x$ .
Using the product rule, the derivative of $x \ln x$ is $1 \cdot \ln x + x \cdot \frac{1}{x}$ .
Using the chain rule, the derivative of $-x$ is $-1$ .
So, the derivative of the numerator is $\ln x +1 -1 = \ln x$ .

STEP 6

Now we can rewrite the limit using the derivatives of the numerator and the denominator.
$\lim{x \rightarrow e}\frac{\ln x}{0}$

STEP 7

This limit is undefined because we cannot divide by zero. However, we can rewrite the limit as the limit of the reciprocal of the function, which will allow us to apply L'Hopital's rule again.
$\lim{x \rightarrow e}\frac{1}{\frac{1}{\ln x}}$

STEP 8

Now, we can apply L'Hopital's rule again. This means we need to take the derivative of the numerator and the derivative of the denominator.
The derivative of the numerator, $1$ , is $0$ .
The derivative of the denominator, $\frac{1}{\ln x}$ , can be found using the chain rule.

STEP 9

Find the derivative of the denominator, $\frac{}{\ln x}$ .
Using the chain rule, the derivative of $\frac{}{\ln x}$ is $-\frac{}{x(\ln x)^2}$ .

STEP 10

Now we can rewrite the limit using the derivatives of the numerator and the denominator.
$\lim{x \rightarrow e}\frac{0}{-\frac{}{x(\ln x)^2}}$

STEP 11

This limit is $0$ because the numerator is $0$ .
So, the solution to the limit problem is $0$ .

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QuestionCalculate the limit: $\lim _{x \rightarrow e}(x \ln x - x)$

Studdy Solution

STEP 1

Assumptions1. We are dealing with a limit problem where $x$ approaches $e$ . . The function is $x \ln x - x$ .
3. We are using the natural logarithm, which has a base of $e$ .

STEP 2

STEP 3

Rewrite the function $x \ln x - x$ as a fraction.
$\lim{x \rightarrow e}\frac{x \ln x - x}{1}$

STEP 4

Now, we can apply L'Hopital's rule. This means we need to take the derivative of the numerator and the derivative of the denominator.
The derivative of the numerator, $x \ln x - x$ , can be found using the product rule and the chain rule. The derivative of the denominator, $1$ , is $0$ .

STEP 5

Find the derivative of the numerator, $x \ln x - x$ .
Using the product rule, the derivative of $x \ln x$ is $1 \cdot \ln x + x \cdot \frac{1}{x}$ .
Using the chain rule, the derivative of $-x$ is $-1$ .
So, the derivative of the numerator is $\ln x +1 -1 = \ln x$ .

STEP 6

Now we can rewrite the limit using the derivatives of the numerator and the denominator.
$\lim{x \rightarrow e}\frac{\ln x}{0}$

STEP 7

This limit is undefined because we cannot divide by zero. However, we can rewrite the limit as the limit of the reciprocal of the function, which will allow us to apply L'Hopital's rule again.
$\lim{x \rightarrow e}\frac{1}{\frac{1}{\ln x}}$

STEP 8

Now, we can apply L'Hopital's rule again. This means we need to take the derivative of the numerator and the derivative of the denominator.
The derivative of the numerator, $1$ , is $0$ .
The derivative of the denominator, $\frac{1}{\ln x}$ , can be found using the chain rule.

STEP 9

Find the derivative of the denominator, $\frac{}{\ln x}$ .
Using the chain rule, the derivative of $\frac{}{\ln x}$ is $-\frac{}{x(\ln x)^2}$ .

STEP 10

Now we can rewrite the limit using the derivatives of the numerator and the denominator.
$\lim{x \rightarrow e}\frac{0}{-\frac{}{x(\ln x)^2}}$

STEP 11

This limit is $0$ because the numerator is $0$ .
So, the solution to the limit problem is $0$ .

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QuestionCalculate the limit: lim⁡x→e(xln⁡x−x)\lim _{x \rightarrow e}(x \ln x - x)limx→e​(xlnx−x)

Studdy Solution

STEP 1

Assumptions1. We are dealing with a limit problem where xxx approaches eee. . The function is xln⁡x−xx \ln x - xxlnx−x.3. We are using the natural logarithm, which has a base of eee.

STEP 2

STEP 3

Rewrite the function xln⁡x−xx \ln x - xxlnx−x as a fraction.lim⁡x→exln⁡x−x1\lim{x \rightarrow e}\frac{x \ln x - x}{1}limx→e1xlnx−x​

STEP 4

STEP 5

STEP 6

Now we can rewrite the limit using the derivatives of the numerator and the denominator.lim⁡x→eln⁡x0\lim{x \rightarrow e}\frac{\ln x}{0}limx→e0lnx​

STEP 7

This limit is undefined because we cannot divide by zero. However, we can rewrite the limit as the limit of the reciprocal of the function, which will allow us to apply L'Hopital's rule again.lim⁡x→e11ln⁡x\lim{x \rightarrow e}\frac{1}{\frac{1}{\ln x}}limx→elnx1​1​

STEP 8

Now, we can apply L'Hopital's rule again. This means we need to take the derivative of the numerator and the derivative of the denominator.The derivative of the numerator, 111, is 000.The derivative of the denominator, 1ln⁡x\frac{1}{\ln x}lnx1​, can be found using the chain rule.

STEP 9

Find the derivative of the denominator, ln⁡x\frac{}{\ln x}lnx​.Using the chain rule, the derivative of ln⁡x\frac{}{\ln x}lnx​ is −x(ln⁡x)2-\frac{}{x(\ln x)^2}−x(lnx)2​.

STEP 10

Now we can rewrite the limit using the derivatives of the numerator and the denominator.lim⁡x→e0−x(ln⁡x)2\lim{x \rightarrow e}\frac{0}{-\frac{}{x(\ln x)^2}}limx→e−x(lnx)2​0​

STEP 11

This limit is 000 because the numerator is 000.So, the solution to the limit problem is 000.

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QuestionCalculate the limit: $\lim _{x \rightarrow e}(x \ln x - x)$

Assumptions1. We are dealing with a limit problem where $x$ approaches $e$ . . The function is $x \ln x - x$ .
3. We are using the natural logarithm, which has a base of $e$ .

Rewrite the function $x \ln x - x$ as a fraction.
$\lim{x \rightarrow e}\frac{x \ln x - x}{1}$

Now we can rewrite the limit using the derivatives of the numerator and the denominator.
$\lim{x \rightarrow e}\frac{\ln x}{0}$

This limit is undefined because we cannot divide by zero. However, we can rewrite the limit as the limit of the reciprocal of the function, which will allow us to apply L'Hopital's rule again.
$\lim{x \rightarrow e}\frac{1}{\frac{1}{\ln x}}$

Now, we can apply L'Hopital's rule again. This means we need to take the derivative of the numerator and the derivative of the denominator.
The derivative of the numerator, $1$ , is $0$ .
The derivative of the denominator, $\frac{1}{\ln x}$ , can be found using the chain rule.

Find the derivative of the denominator, $\frac{}{\ln x}$ .
Using the chain rule, the derivative of $\frac{}{\ln x}$ is $-\frac{}{x(\ln x)^2}$ .

Now we can rewrite the limit using the derivatives of the numerator and the denominator.
$\lim{x \rightarrow e}\frac{0}{-\frac{}{x(\ln x)^2}}$

This limit is $0$ because the numerator is $0$ .
So, the solution to the limit problem is $0$ .