Calculus Examples

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

Step 1

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $θ$ .

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

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.

$\frac{1}{2} \cdot \frac{lim_{θ \to 0} 1 - \cos (θ)}{lim_{θ \to 0} \sin^{2} (θ)}$

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 $θ$ approaches $0$ .

$\frac{1}{2} \cdot \frac{lim_{θ \to 0} 1 - lim_{θ \to 0} \cos (θ)}{lim_{θ \to 0} \sin^{2} (θ)}$

Step 2.1.2.1.2

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

$\frac{1}{2} \cdot \frac{1 - lim_{θ \to 0} \cos (θ)}{lim_{θ \to 0} \sin^{2} (θ)}$

Step 2.1.2.1.3

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

$\frac{1}{2} \cdot \frac{1 - \cos (lim_{θ \to 0} θ)}{lim_{θ \to 0} \sin^{2} (θ)}$

Step 2.1.2.2

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

$\frac{1}{2} \cdot \frac{1 - \cos (0)}{lim_{θ \to 0} \sin^{2} (θ)}$

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

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

$\frac{1}{2} \cdot \frac{1 - 1 \cdot 1}{lim_{θ \to 0} \sin^{2} (θ)}$

Step 2.1.2.3.1.2

Multiply $- 1$ by $1$ .

$\frac{1}{2} \cdot \frac{1 - 1}{lim_{θ \to 0} \sin^{2} (θ)}$

Step 2.1.2.3.2

Subtract $1$ from $1$ .

$\frac{1}{2} \cdot \frac{0}{lim_{θ \to 0} \sin^{2} (θ)}$

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

Move the exponent $2$ from $\sin^{2} (θ)$ outside the limit using the Limits Power Rule.

$\frac{1}{2} \cdot \frac{0}{{(lim_{θ \to 0} \sin (θ))}^{2}}$

Step 2.1.3.1.2

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

$\frac{1}{2} \cdot \frac{0}{\sin^{2} (lim_{θ \to 0} θ)}$

Step 2.1.3.2

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

$\frac{1}{2} \cdot \frac{0}{\sin^{2} (0)}$

Step 2.1.3.3

Simplify the answer.

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Step 2.1.3.3.1

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

$\frac{1}{2} \cdot \frac{0}{0^{2}}$

Step 2.1.3.3.2

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

$\frac{1}{2} \cdot \frac{0}{0}$

Step 2.1.3.3.3

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

Undefined

$\frac{1}{2} \cdot \frac{0}{0}$

Step 2.1.3.4

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

Undefined

$\frac{1}{2} \cdot \frac{0}{0}$

Step 2.1.4

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

Undefined

$\frac{1}{2} \cdot \frac{0}{0}$

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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Step 2.3.4.1

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

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

Step 2.3.4.2

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

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

Step 2.3.4.3

Multiply $- 1$ by $- 1$ .

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

Step 2.3.4.4

Multiply $\sin (θ)$ by $1$ .

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

Step 2.3.5

Add $0$ and $\sin (θ)$ .

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

Step 2.3.6

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

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Step 2.3.6.1

To apply the Chain Rule, set $u$ as $\sin (θ)$ .

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

Step 2.3.6.2

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

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

Step 2.3.6.3

Replace all occurrences of $u$ with $\sin (θ)$ .

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

Step 2.3.7

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

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

Step 2.3.8

Simplify.

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Step 2.3.8.1

Reorder the factors of $2 \sin (θ) \cos (θ)$ .

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

Step 2.3.8.2

Reorder $2 \cos (θ)$ and $\sin (θ)$ .

$\frac{1}{2} lim_{θ \to 0} \frac{\sin (θ)}{\sin (θ) \cdot 2 \cos (θ)}$

Step 2.3.8.3

Reorder $\sin (θ)$ and $2$ .

$\frac{1}{2} lim_{θ \to 0} \frac{\sin (θ)}{2 \cdot \sin (θ) \cos (θ)}$

Step 2.3.8.4

Apply the sine double-angle identity.

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

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.

$\frac{1}{2} \cdot \frac{lim_{θ \to 0} \sin (θ)}{lim_{θ \to 0} \sin (2 θ)}$

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

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

$\frac{1}{2} \cdot \frac{\sin (lim_{θ \to 0} θ)}{lim_{θ \to 0} \sin (2 θ)}$

Step 3.1.2.2

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

$\frac{1}{2} \cdot \frac{\sin (0)}{lim_{θ \to 0} \sin (2 θ)}$

Step 3.1.2.3

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

$\frac{1}{2} \cdot \frac{0}{lim_{θ \to 0} \sin (2 θ)}$

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

Evaluate the limit.

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Step 3.1.3.1.1

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

$\frac{1}{2} \cdot \frac{0}{\sin (lim_{θ \to 0} 2 θ)}$

Step 3.1.3.1.2

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

$\frac{1}{2} \cdot \frac{0}{\sin (2 lim_{θ \to 0} θ)}$

Step 3.1.3.2

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

$\frac{1}{2} \cdot \frac{0}{\sin (2 \cdot 0)}$

Step 3.1.3.3

Simplify the answer.

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Step 3.1.3.3.1

Multiply $2$ by $0$ .

$\frac{1}{2} \cdot \frac{0}{\sin (0)}$

Step 3.1.3.3.2

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

$\frac{1}{2} \cdot \frac{0}{0}$

Step 3.1.3.3.3

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

Undefined

$\frac{1}{2} \cdot \frac{0}{0}$

Step 3.1.3.4

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

Undefined

$\frac{1}{2} \cdot \frac{0}{0}$

Step 3.1.4

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

Undefined

$\frac{1}{2} \cdot \frac{0}{0}$

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.

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

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

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

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

Step 3.3.3

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

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Step 3.3.3.1

To apply the Chain Rule, set $u$ as $2 θ$ .

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

Step 3.3.3.2

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

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

Step 3.3.3.3

Replace all occurrences of $u$ with $2 θ$ .

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

Step 3.3.4

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

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

Step 3.3.5

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

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

Step 3.3.6

Multiply $2$ by $1$ .

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

Step 3.3.7

Move $2$ to the left of $\cos (2 θ)$ .

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

Step 4

Evaluate the limit.

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Step 4.1

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $θ$ .

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 6

Simplify the answer.

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Step 6.1

Multiply $\frac{1}{2} \cdot \frac{1}{2}$ .

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Step 6.1.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

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

Step 6.1.2

Multiply $2$ by $2$ .

$\frac{1}{4} \cdot \frac{\cos (0)}{\cos (2 \cdot 0)}$

Step 6.2

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

$\frac{1}{4} \cdot \frac{1}{\cos (2 \cdot 0)}$

Step 6.3

Simplify the denominator.

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Step 6.3.1

Multiply $2$ by $0$ .

$\frac{1}{4} \cdot \frac{1}{\cos (0)}$

Step 6.3.2

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

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

Step 6.4

Cancel the common factor of $1$ .

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Step 6.4.1

Cancel the common factor.

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

Step 6.4.2

Rewrite the expression.

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

Step 6.5

Multiply $\frac{1}{4}$ by $1$ .

$\frac{1}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{4}$

Decimal Form:

$0.25$