Calculus Examples

$lim_{x \to 1} \frac{x^{2} - 1 + \ln (x)}{e^{x} - e}$

Step 1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to 1} x^{2} - 1 + \ln (x)}{lim_{x \to 1} e^{x} - e}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{lim_{x \to 1} x^{2} - lim_{x \to 1} 1 + lim_{x \to 1} \ln (x)}{lim_{x \to 1} e^{x} - e}$

Step 1.2.2

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 1} x)}^{2} - lim_{x \to 1} 1 + lim_{x \to 1} \ln (x)}{lim_{x \to 1} e^{x} - e}$

Step 1.2.3

Evaluate the limit of $1$ which is constant as $x$ approaches $1$ .

$\frac{{(lim_{x \to 1} x)}^{2} - 1 \cdot 1 + lim_{x \to 1} \ln (x)}{lim_{x \to 1} e^{x} - e}$

Step 1.2.4

Move the limit inside the logarithm.

$\frac{{(lim_{x \to 1} x)}^{2} - 1 \cdot 1 + \ln (lim_{x \to 1} x)}{lim_{x \to 1} e^{x} - e}$

Step 1.2.5

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 1.2.5.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{1^{2} - 1 \cdot 1 + \ln (lim_{x \to 1} x)}{lim_{x \to 1} e^{x} - e}$

Step 1.2.5.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{1^{2} - 1 \cdot 1 + \ln (1)}{lim_{x \to 1} e^{x} - e}$

Step 1.2.6

Simplify the answer.

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Step 1.2.6.1

Simplify each term.

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Step 1.2.6.1.1

One to any power is one.

$\frac{1 - 1 \cdot 1 + \ln (1)}{lim_{x \to 1} e^{x} - e}$

Step 1.2.6.1.2

Multiply $- 1$ by $1$ .

$\frac{1 - 1 + \ln (1)}{lim_{x \to 1} e^{x} - e}$

Step 1.2.6.1.3

The natural logarithm of $1$ is $0$ .

$\frac{1 - 1 + 0}{lim_{x \to 1} e^{x} - e}$

Step 1.2.6.2

Subtract $1$ from $1$ .

$\frac{0 + 0}{lim_{x \to 1} e^{x} - e}$

Step 1.2.6.3

Add $0$ and $0$ .

$\frac{0}{lim_{x \to 1} e^{x} - e}$

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Evaluate the limit.

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Step 1.3.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{0}{lim_{x \to 1} e^{x} - lim_{x \to 1} e}$

Step 1.3.1.2

Move the limit into the exponent.

$\frac{0}{e^{lim_{x \to 1} x} - lim_{x \to 1} e}$

Step 1.3.1.3

Evaluate the limit of $e$ which is constant as $x$ approaches $1$ .

$\frac{0}{e^{lim_{x \to 1} x} - e}$

Step 1.3.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{0}{e^{1} - e}$

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Simplify.

$\frac{0}{e - e}$

Step 1.3.3.2

Subtract $e$ from $e$ .

$\frac{0}{0}$

Step 1.3.3.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.3.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 1} \frac{x^{2} - 1 + \ln (x)}{e^{x} - e} = lim_{x \to 1} \frac{\frac{d}{d x} [x^{2} - 1 + \ln (x)]}{\frac{d}{d x} [e^{x} - e]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 1} \frac{\frac{d}{d x} [x^{2} - 1 + \ln (x)]}{\frac{d}{d x} [e^{x} - e]}$

Step 3.2

By the Sum Rule, the derivative of $x^{2} - 1 + \ln (x)$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1] + \frac{d}{d x} [\ln (x)]$ .

$lim_{x \to 1} \frac{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1] + \frac{d}{d x} [\ln (x)]}{\frac{d}{d x} [e^{x} - e]}$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 1} \frac{2 x + \frac{d}{d x} [- 1] + \frac{d}{d x} [\ln (x)]}{\frac{d}{d x} [e^{x} - e]}$

Step 3.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$lim_{x \to 1} \frac{2 x + 0 + \frac{d}{d x} [\ln (x)]}{\frac{d}{d x} [e^{x} - e]}$

Step 3.5

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$lim_{x \to 1} \frac{2 x + 0 + \frac{1}{x}}{\frac{d}{d x} [e^{x} - e]}$

Step 3.6

Add $2 x$ and $0$ .

$lim_{x \to 1} \frac{2 x + \frac{1}{x}}{\frac{d}{d x} [e^{x} - e]}$

Step 3.7

By the Sum Rule, the derivative of $e^{x} - e$ with respect to $x$ is $\frac{d}{d x} [e^{x}] + \frac{d}{d x} [- e]$ .

$lim_{x \to 1} \frac{2 x + \frac{1}{x}}{\frac{d}{d x} [e^{x}] + \frac{d}{d x} [- e]}$

Step 3.8

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$lim_{x \to 1} \frac{2 x + \frac{1}{x}}{e^{x} + \frac{d}{d x} [- e]}$

Step 3.9

Since $- e$ is constant with respect to $x$ , the derivative of $- e$ with respect to $x$ is $0$ .

$lim_{x \to 1} \frac{2 x + \frac{1}{x}}{e^{x} + 0}$

Step 3.10

Add $e^{x}$ and $0$ .

$lim_{x \to 1} \frac{2 x + \frac{1}{x}}{e^{x}}$

Step 4

Combine terms.

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Step 4.1

To write $2 x$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$lim_{x \to 1} \frac{\frac{2 x \cdot x}{x} + \frac{1}{x}}{e^{x}}$

Step 4.2

Combine the numerators over the common denominator.

$lim_{x \to 1} \frac{\frac{2 x \cdot x + 1}{x}}{e^{x}}$

Step 5

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $1$ .

$\frac{lim_{x \to 1} \frac{2 x \cdot x + 1}{x}}{lim_{x \to 1} e^{x}}$

Step 6

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $1$ .

$\frac{\frac{lim_{x \to 1} 2 x \cdot x + 1}{lim_{x \to 1} x}}{lim_{x \to 1} e^{x}}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{\frac{lim_{x \to 1} 2 x \cdot x + lim_{x \to 1} 1}{lim_{x \to 1} x}}{lim_{x \to 1} e^{x}}$

Step 8

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{\frac{2 lim_{x \to 1} x \cdot x + lim_{x \to 1} 1}{lim_{x \to 1} x}}{lim_{x \to 1} e^{x}}$

Step 9

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{\frac{2 lim_{x \to 1} x \cdot lim_{x \to 1} x + lim_{x \to 1} 1}{lim_{x \to 1} x}}{lim_{x \to 1} e^{x}}$

Step 10

Evaluate the limit of $1$ which is constant as $x$ approaches $1$ .

$\frac{\frac{2 lim_{x \to 1} x \cdot lim_{x \to 1} x + 1}{lim_{x \to 1} x}}{lim_{x \to 1} e^{x}}$

Step 11

Move the limit into the exponent.

$\frac{\frac{2 lim_{x \to 1} x \cdot lim_{x \to 1} x + 1}{lim_{x \to 1} x}}{e^{lim_{x \to 1} x}}$

Step 12

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 12.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{\frac{2 \cdot 1 \cdot lim_{x \to 1} x + 1}{lim_{x \to 1} x}}{e^{lim_{x \to 1} x}}$

Step 12.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{\frac{2 \cdot 1 \cdot 1 + 1}{lim_{x \to 1} x}}{e^{lim_{x \to 1} x}}$

Step 12.3

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{\frac{2 \cdot 1 \cdot 1 + 1}{1}}{e^{lim_{x \to 1} x}}$

Step 12.4

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{\frac{2 \cdot 1 \cdot 1 + 1}{1}}{e^{1}}$

Step 13

Simplify the answer.

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Step 13.1

Divide $2 \cdot 1 \cdot 1 + 1$ by $1$ .

$\frac{2 \cdot 1 \cdot 1 + 1}{e^{1}}$

Step 13.2

Simplify the numerator.

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Step 13.2.1

Multiply $2 \cdot 1 \cdot 1$ .

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Step 13.2.1.1

Multiply $2$ by $1$ .

$\frac{2 \cdot 1 + 1}{e^{1}}$

Step 13.2.1.2

Multiply $2$ by $1$ .

$\frac{2 + 1}{e^{1}}$

Step 13.2.2

Add $2$ and $1$ .

$\frac{3}{e^{1}}$

Step 13.3

Simplify.

$\frac{3}{e}$