Calculus Examples

$lim_{x \to 1} \frac{4 (x^{2} - 1)}{x - 1}$

Step 1

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$4 lim_{x \to 1} \frac{x^{2} - 1}{x - 1}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

$4 \frac{lim_{x \to 1} x^{2} - 1}{lim_{x \to 1} x - 1}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Evaluate the limit.

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Step 2.1.2.1.1

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

$4 \frac{lim_{x \to 1} x^{2} - lim_{x \to 1} 1}{lim_{x \to 1} x - 1}$

Step 2.1.2.1.2

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

$4 \frac{{(lim_{x \to 1} x)}^{2} - lim_{x \to 1} 1}{lim_{x \to 1} x - 1}$

Step 2.1.2.1.3

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

$4 \frac{{(lim_{x \to 1} x)}^{2} - 1 \cdot 1}{lim_{x \to 1} x - 1}$

Step 2.1.2.2

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

$4 \frac{1^{2} - 1 \cdot 1}{lim_{x \to 1} x - 1}$

Step 2.1.2.3

Simplify the answer.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

One to any power is one.

$4 \frac{1 - 1 \cdot 1}{lim_{x \to 1} x - 1}$

Step 2.1.2.3.1.2

Multiply $- 1$ by $1$ .

$4 \frac{1 - 1}{lim_{x \to 1} x - 1}$

Step 2.1.2.3.2

Subtract $1$ from $1$ .

$4 \frac{0}{lim_{x \to 1} x - 1}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

Evaluate the limit.

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Step 2.1.3.1.1

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

$4 \frac{0}{lim_{x \to 1} x - lim_{x \to 1} 1}$

Step 2.1.3.1.2

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

$4 \frac{0}{lim_{x \to 1} x - 1 \cdot 1}$

Step 2.1.3.2

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

$4 \frac{0}{1 - 1 \cdot 1}$

Step 2.1.3.3

Simplify the answer.

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Step 2.1.3.3.1

Multiply $- 1$ by $1$ .

$4 \frac{0}{1 - 1}$

Step 2.1.3.3.2

Subtract $1$ from $1$ .

$4 (\frac{0}{0})$

Step 2.1.3.3.3

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

Undefined

$4 (\frac{0}{0})$

Step 2.1.3.4

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

Undefined

$4 (\frac{0}{0})$

Step 2.1.4

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

Undefined

$4 (\frac{0}{0})$

Step 2.2

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$4 lim_{x \to 1} \frac{\frac{d}{d x} [x^{2} - 1]}{\frac{d}{d x} [x - 1]}$

Step 2.3.2

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

$4 lim_{x \to 1} \frac{\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]}{\frac{d}{d x} [x - 1]}$

Step 2.3.3

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

$4 lim_{x \to 1} \frac{2 x + \frac{d}{d x} [- 1]}{\frac{d}{d x} [x - 1]}$

Step 2.3.4

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

$4 lim_{x \to 1} \frac{2 x + 0}{\frac{d}{d x} [x - 1]}$

Step 2.3.5

Add $2 x$ and $0$ .

$4 lim_{x \to 1} \frac{2 x}{\frac{d}{d x} [x - 1]}$

Step 2.3.6

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

$4 lim_{x \to 1} \frac{2 x}{\frac{d}{d x} [x] + \frac{d}{d x} [- 1]}$

Step 2.3.7

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

$4 lim_{x \to 1} \frac{2 x}{1 + \frac{d}{d x} [- 1]}$

Step 2.3.8

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

$4 lim_{x \to 1} \frac{2 x}{1 + 0}$

Step 2.3.9

Add $1$ and $0$ .

$4 lim_{x \to 1} \frac{2 x}{1}$

Step 2.4

Divide $2 x$ by $1$ .

$4 lim_{x \to 1} 2 x$

Step 3

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

$4 \cdot 2 lim_{x \to 1} x$

Step 4

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

$4 \cdot 2 \cdot 1$

Step 5

Multiply $4 \cdot 2 \cdot 1$ .

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Step 5.1

Multiply $4$ by $2$ .

$8 \cdot 1$

Step 5.2

Multiply $8$ by $1$ .

$8$