Calculus Examples

$lim_{x \to 0} \frac{\sin (x^{2})}{x}$

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.

$\frac{lim_{x \to 0} \sin (x^{2})}{lim_{x \to 0} x}$

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.

$\frac{\sin (lim_{x \to 0} x^{2})}{lim_{x \to 0} x}$

Step 1.2.1.2

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

$\frac{\sin ({(lim_{x \to 0} x)}^{2})}{lim_{x \to 0} x}$

Step 1.2.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\sin (0^{2})}{lim_{x \to 0} x}$

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

Raising $0$ to any positive power yields $0$ .

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

Step 1.2.3.2

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

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

Step 1.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0}{0}$

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.

$lim_{x \to 0} \frac{\sin (x^{2})}{x} = lim_{x \to 0} \frac{\frac{d}{d x} [\sin (x^{2})]}{\frac{d}{d x} [x]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [\sin (x^{2})]}{\frac{d}{d x} [x]}$

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 $g (x) = x^{2}$ .

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Step 3.2.1

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

$lim_{x \to 0} \frac{\frac{d}{d u} [\sin (u)] \frac{d}{d x} [x^{2}]}{\frac{d}{d x} [x]}$

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 4

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

$lim_{x \to 0} 2 x \cos (x^{2})$

Step 5

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

$2 lim_{x \to 0} x \cos (x^{2})$

Step 6

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$2 lim_{x \to 0} x \cdot lim_{x \to 0} \cos (x^{2})$

Step 7

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

$2 lim_{x \to 0} x \cdot \cos (lim_{x \to 0} x^{2})$

Step 8

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

$2 lim_{x \to 0} x \cdot \cos ({(lim_{x \to 0} x)}^{2})$

Step 9

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$2 \cdot 0 \cdot \cos ({(lim_{x \to 0} x)}^{2})$

Step 9.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$2 \cdot 0 \cdot \cos (0^{2})$

Step 10

Simplify the answer.

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Step 10.1

Multiply $2$ by $0$ .

$0 \cdot \cos (0^{2})$

Step 10.2

Raising $0$ to any positive power yields $0$ .

$0 \cdot \cos (0)$

Step 10.3

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

$0 \cdot 1$

Step 10.4

Multiply $0$ by $1$ .

$0$