Calculus Examples

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

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.

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

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Evaluate the limit.

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Step 1.2.1.1

Move the limit inside the trig function because sine is continuous.

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

Step 1.2.1.2

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

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

Step 1.2.1.3

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

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

Step 1.2.1.4

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

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

Step 1.2.2

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

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

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

One to any power is one.

$\frac{\sin (1 - 1 \cdot 1)}{lim_{x \to 1} x - 1}$

Step 1.2.3.1.2

Multiply $- 1$ by $1$ .

$\frac{\sin (1 - 1)}{lim_{x \to 1} x - 1}$

Step 1.2.3.2

Subtract $1$ from $1$ .

$\frac{\sin (0)}{lim_{x \to 1} x - 1}$

Step 1.2.3.3

The exact value of $\sin (0)$ is $0$ .

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

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 $1$ .

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

Step 1.3.1.2

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

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

Step 1.3.2

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

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

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Multiply $- 1$ by $1$ .

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

Step 1.3.3.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 1 - x^{2}$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $1 - x^{2}$ .

$lim_{x \to 1} \frac{\frac{d}{d u} [\sin (u)] \frac{d}{d x} [1 - x^{2}]}{\frac{d}{d x} [x - 1]}$

Step 3.2.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$lim_{x \to 1} \frac{\cos (u) \frac{d}{d x} [1 - x^{2}]}{\frac{d}{d x} [x - 1]}$

Step 3.2.3

Replace all occurrences of $u$ with $1 - x^{2}$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

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

Step 3.6

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

Step 3.7

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

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

Step 3.8

Multiply $2$ by $- 1$ .

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

Step 3.9

Reorder the factors of $\cos (1 - x^{2}) (- 2 x)$ .

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

Step 3.10

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

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

Step 3.11

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

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

Step 3.12

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

$lim_{x \to 1} \frac{- 2 x \cos (1 - x^{2})}{1 + 0}$

Step 3.13

Add $1$ and $0$ .

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

Step 4

Divide $- 2 x \cos (1 - x^{2})$ by $1$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Move the limit inside the trig function because cosine is continuous.

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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Step 11.1

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

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

Step 11.2

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

$- 2 \cdot 1 \cdot \cos (1 - 1^{2})$

Step 12

Simplify the answer.

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Step 12.1

Multiply $- 2$ by $1$ .

$- 2 \cdot \cos (1 - 1^{2})$

Step 12.2

Simplify each term.

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Step 12.2.1

One to any power is one.

$- 2 \cdot \cos (1 - 1 \cdot 1)$

Step 12.2.2

Multiply $- 1$ by $1$ .

$- 2 \cdot \cos (1 - 1)$

Step 12.3

Subtract $1$ from $1$ .

$- 2 \cdot \cos (0)$

Step 12.4

The exact value of $\cos (0)$ is $1$ .

$- 2 \cdot 1$

Step 12.5

Multiply $- 2$ by $1$ .

$- 2$