Calculus Examples

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

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

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

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

Step 1.1.3.1.2

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

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

Step 1.1.3.1.3

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

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

Step 1.1.3.2

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

$\frac{0}{1 - \cos (0)}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

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

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

Step 1.1.3.3.1.2

Multiply $- 1$ by $1$ .

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

Step 1.1.3.3.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

Step 1.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.1.4

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

Undefined

$\frac{0}{0}$

Step 1.2

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

Evaluate $\frac{d}{d x} [- \cos (x)]$ .

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Step 1.3.5.1

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

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

Step 1.3.5.2

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

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

Step 1.3.5.3

Multiply $- 1$ by $- 1$ .

$lim_{x \to 0} \frac{2 x}{0 + 1 \sin (x)}$

Step 1.3.5.4

Multiply $\sin (x)$ by $1$ .

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

Step 1.3.6

Add $0$ and $\sin (x)$ .

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

Step 2

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

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

Step 3

Apply L'Hospital's rule.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

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

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

Step 3.1.3.2

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

$2 \frac{0}{\sin (0)}$

Step 3.1.3.3

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

$2 (\frac{0}{0})$

Step 3.1.3.4

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

Undefined

$2 (\frac{0}{0})$

Step 3.1.4

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

Undefined

$2 (\frac{0}{0})$

Step 3.2

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.

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

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

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

$2 lim_{x \to 0} \frac{1}{\frac{d}{d x} [\sin (x)]}$

Step 3.3.3

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

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

Step 4

Evaluate the limit.

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Step 4.1

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

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

Step 4.2

Evaluate the limit of $1$ which is constant as $x$ approaches $0$ .

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

Step 4.3

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

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

Step 5

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

$2 \frac{1}{\cos (0)}$

Step 6

Simplify the answer.

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Step 6.1

Convert from $\frac{1}{\cos (0)}$ to $\sec (0)$ .

$2 \sec (0)$

Step 6.2

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

$2 \cdot 1$

Step 6.3

Multiply $2$ by $1$ .

$2$