Calculus Examples

$lim_{θ \to \frac{π}{2}} \tan^{2} (θ) (1 - \sin (θ))$

Step 1

Rewrite $\tan^{2} (θ) (1 - \sin (θ))$ as $\frac{1 - \sin (θ)}{\tan^{- 2} (θ)}$ .

$lim_{θ \to \frac{π}{2}} \frac{1 - \sin (θ)}{\tan^{- 2} (θ)}$

Step 2

Set up the limit as a left-sided limit.

$lim_{θ \to {(\frac{π}{2})}^{-}} \frac{1 - \sin (θ)}{\tan^{- 2} (θ)}$

Step 3

Evaluate the left-sided limit.

Tap for more steps...

Step 3.1

Apply L'Hospital's rule.

Tap for more steps...

Step 3.1.1

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

Tap for more steps...

Step 3.1.1.1

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

$\frac{lim_{θ \to {(\frac{π}{2})}^{-}} 1 - \sin (θ)}{lim_{θ \to {(\frac{π}{2})}^{-}} \tan^{- 2} (θ)}$

Step 3.1.1.2

Evaluate the limit of the numerator.

Tap for more steps...

Step 3.1.1.2.1

Evaluate the limit.

Tap for more steps...

Step 3.1.1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as $θ$ approaches $\frac{π}{2}$ .

$\frac{lim_{θ \to {(\frac{π}{2})}^{-}} 1 - lim_{θ \to {(\frac{π}{2})}^{-}} \sin (θ)}{lim_{θ \to {(\frac{π}{2})}^{-}} \tan^{- 2} (θ)}$

Step 3.1.1.2.1.2

Evaluate the limit of $1$ which is constant as $θ$ approaches $\frac{π}{2}$ .

$\frac{1 - lim_{θ \to {(\frac{π}{2})}^{-}} \sin (θ)}{lim_{θ \to {(\frac{π}{2})}^{-}} \tan^{- 2} (θ)}$

Step 3.1.1.2.1.3

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

$\frac{1 - \sin (lim_{θ \to {(\frac{π}{2})}^{-}} θ)}{lim_{θ \to {(\frac{π}{2})}^{-}} \tan^{- 2} (θ)}$

Step 3.1.1.2.2

Evaluate the limit of $θ$ by plugging in $\frac{π}{2}$ for $θ$ .

$\frac{1 - \sin (\frac{π}{2})}{lim_{θ \to {(\frac{π}{2})}^{-}} \tan^{- 2} (θ)}$

Step 3.1.1.2.3

Simplify the answer.

Tap for more steps...

Step 3.1.1.2.3.1

Simplify each term.

Tap for more steps...

Step 3.1.1.2.3.1.1

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$\frac{1 - 1 \cdot 1}{lim_{θ \to {(\frac{π}{2})}^{-}} \tan^{- 2} (θ)}$

Step 3.1.1.2.3.1.2

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{lim_{θ \to {(\frac{π}{2})}^{-}} \tan^{- 2} (θ)}$

Step 3.1.1.2.3.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{θ \to {(\frac{π}{2})}^{-}} \tan^{- 2} (θ)}$

Step 3.1.1.3

Evaluate the limit of the denominator.

Tap for more steps...

Step 3.1.1.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{0}{lim_{θ \to {(\frac{π}{2})}^{-}} \frac{1}{\tan^{2} (θ)}}$

Step 3.1.1.3.2

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{\tan^{2} (θ)}$ approaches $0$ .

$\frac{0}{0}$

Step 3.1.1.3.3

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

Undefined

$\frac{0}{0}$

Step 3.1.1.4

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

Undefined

$\frac{0}{0}$

Step 3.1.2

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.

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

Step 3.1.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 3.1.3.1

Differentiate the numerator and denominator.

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

Step 3.1.3.2

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

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

Step 3.1.3.3

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

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

Step 3.1.3.4

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

Tap for more steps...

Step 3.1.3.4.1

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

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

Step 3.1.3.4.2

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

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

Step 3.1.3.5

Subtract $\cos (θ)$ from $0$ .

$lim_{θ \to {(\frac{π}{2})}^{-}} \frac{- \cos (θ)}{\frac{d}{d θ} [\tan^{- 2} (θ)]}$

Step 3.1.3.6

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 $g (θ) = \tan (θ)$ .

Tap for more steps...

Step 3.1.3.6.1

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

$lim_{θ \to {(\frac{π}{2})}^{-}} \frac{- \cos (θ)}{\frac{d}{d u} [u^{- 2}] \frac{d}{d θ} [\tan (θ)]}$

Step 3.1.3.6.2

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

$lim_{θ \to {(\frac{π}{2})}^{-}} \frac{- \cos (θ)}{- 2 u^{- 3} \frac{d}{d θ} [\tan (θ)]}$

Step 3.1.3.6.3

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

$lim_{θ \to {(\frac{π}{2})}^{-}} \frac{- \cos (θ)}{- 2 \tan^{- 3} (θ) \frac{d}{d θ} [\tan (θ)]}$

Step 3.1.3.7

The derivative of $\tan (θ)$ with respect to $θ$ is $\sec^{2} (θ)$ .

$lim_{θ \to {(\frac{π}{2})}^{-}} \frac{- \cos (θ)}{- 2 \tan^{- 3} (θ) \sec^{2} (θ)}$

Step 3.1.3.8

Simplify.

Tap for more steps...

Step 3.1.3.8.1

Reorder the factors of $- 2 \tan^{- 3} (θ) \sec^{2} (θ)$ .

$lim_{θ \to {(\frac{π}{2})}^{-}} \frac{- \cos (θ)}{- 2 \sec^{2} (θ) \tan^{- 3} (θ)}$

Step 3.1.3.8.2

Rewrite $\sec (θ)$ in terms of sines and cosines.

$lim_{θ \to {(\frac{π}{2})}^{-}} \frac{- \cos (θ)}{- 2 {(\frac{1}{\cos (θ)})}^{2} \tan^{- 3} (θ)}$

Step 3.1.3.8.3

Apply the product rule to $\frac{1}{\cos (θ)}$ .

$lim_{θ \to {(\frac{π}{2})}^{-}} \frac{- \cos (θ)}{- 2 \frac{1^{2}}{\cos^{2} (θ)} \tan^{- 3} (θ)}$

Step 3.1.3.8.4

One to any power is one.

$lim_{θ \to {(\frac{π}{2})}^{-}} \frac{- \cos (θ)}{- 2 \frac{1}{\cos^{2} (θ)} \tan^{- 3} (θ)}$

Step 3.1.3.8.5

Combine $- 2$ and $\frac{1}{\cos^{2} (θ)}$ .

$lim_{θ \to {(\frac{π}{2})}^{-}} \frac{- \cos (θ)}{\frac{- 2}{\cos^{2} (θ)} \tan^{- 3} (θ)}$

Step 3.1.3.8.6

Move the negative in front of the fraction.

$lim_{θ \to {(\frac{π}{2})}^{-}} \frac{- \cos (θ)}{- \frac{2}{\cos^{2} (θ)} \tan^{- 3} (θ)}$

Step 3.1.3.8.7

Rewrite $\tan (θ)$ in terms of sines and cosines.

$lim_{θ \to {(\frac{π}{2})}^{-}} \frac{- \cos (θ)}{- \frac{2}{\cos^{2} (θ)} {(\frac{\sin (θ)}{\cos (θ)})}^{- 3}}$

Step 3.1.3.8.8

Change the sign of the exponent by rewriting the base as its reciprocal.

$lim_{θ \to {(\frac{π}{2})}^{-}} \frac{- \cos (θ)}{- \frac{2}{\cos^{2} (θ)} {(\frac{\cos (θ)}{\sin (θ)})}^{3}}$

Step 3.1.3.8.9

Apply the product rule to $\frac{\cos (θ)}{\sin (θ)}$ .

$lim_{θ \to {(\frac{π}{2})}^{-}} \frac{- \cos (θ)}{- \frac{2}{\cos^{2} (θ)} \cdot \frac{\cos^{3} (θ)}{\sin^{3} (θ)}}$

Step 3.1.3.8.10

Cancel the common factor of $\cos^{2} (θ)$ .

Tap for more steps...

Step 3.1.3.8.10.1

Move the leading negative in $- \frac{2}{\cos^{2} (θ)}$ into the numerator.

$lim_{θ \to {(\frac{π}{2})}^{-}} \frac{- \cos (θ)}{\frac{- 2}{\cos^{2} (θ)} \cdot \frac{\cos^{3} (θ)}{\sin^{3} (θ)}}$

Step 3.1.3.8.10.2

Factor $\cos^{2} (θ)$ out of $\cos^{3} (θ)$ .

$lim_{θ \to {(\frac{π}{2})}^{-}} \frac{- \cos (θ)}{\frac{- 2}{\cos^{2} (θ)} \cdot \frac{\cos^{2} (θ) \cos (θ)}{\sin^{3} (θ)}}$

Step 3.1.3.8.10.3

Cancel the common factor.

$lim_{θ \to {(\frac{π}{2})}^{-}} \frac{- \cos (θ)}{\frac{- 2}{\cos^{2} (θ)} \cdot \frac{\cos^{2} (θ) \cos (θ)}{\sin^{3} (θ)}}$

Step 3.1.3.8.10.4

Rewrite the expression.

$lim_{θ \to {(\frac{π}{2})}^{-}} \frac{- \cos (θ)}{- 2 \frac{\cos (θ)}{\sin^{3} (θ)}}$

Step 3.1.3.8.11

Combine $- 2$ and $\frac{\cos (θ)}{\sin^{3} (θ)}$ .

$lim_{θ \to {(\frac{π}{2})}^{-}} \frac{- \cos (θ)}{\frac{- 2 \cos (θ)}{\sin^{3} (θ)}}$

Step 3.1.3.8.12

Move the negative in front of the fraction.

$lim_{θ \to {(\frac{π}{2})}^{-}} \frac{- \cos (θ)}{- \frac{2 \cos (θ)}{\sin^{3} (θ)}}$

Step 3.1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{θ \to {(\frac{π}{2})}^{-}} - \cos (θ) (- \frac{\sin^{3} (θ)}{2 \cos (θ)})$

Step 3.1.5

Combine factors.

Tap for more steps...

Step 3.1.5.1

Multiply $- 1$ by $- 1$ .

$lim_{θ \to {(\frac{π}{2})}^{-}} 1 \cos (θ) \frac{\sin^{3} (θ)}{2 \cos (θ)}$

Step 3.1.5.2

Multiply $\cos (θ)$ by $1$ .

$lim_{θ \to {(\frac{π}{2})}^{-}} \cos (θ) \frac{\sin^{3} (θ)}{2 \cos (θ)}$

Step 3.1.5.3

Combine $\cos (θ)$ and $\frac{\sin^{3} (θ)}{2 \cos (θ)}$ .

$lim_{θ \to {(\frac{π}{2})}^{-}} \frac{\cos (θ) \sin^{3} (θ)}{2 \cos (θ)}$

Step 3.1.6

Cancel the common factor of $\cos (θ)$ .

Tap for more steps...

Step 3.1.6.1

Cancel the common factor.

$lim_{θ \to {(\frac{π}{2})}^{-}} \frac{\cos (θ) \sin^{3} (θ)}{2 \cos (θ)}$

Step 3.1.6.2

Rewrite the expression.

$lim_{θ \to {(\frac{π}{2})}^{-}} \frac{\sin^{3} (θ)}{2}$

Step 3.2

Evaluate the limit.

Tap for more steps...

Step 3.2.1

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

$\frac{1}{2} lim_{θ \to {(\frac{π}{2})}^{-}} \sin^{3} (θ)$

Step 3.2.2

Move the exponent $3$ from $\sin^{3} (θ)$ outside the limit using the Limits Power Rule.

$\frac{1}{2} {(lim_{θ \to {(\frac{π}{2})}^{-}} \sin (θ))}^{3}$

Step 3.2.3

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

$\frac{1}{2} \sin^{3} (lim_{θ \to {(\frac{π}{2})}^{-}} θ)$

Step 3.3

Evaluate the limit of $θ$ by plugging in $\frac{π}{2}$ for $θ$ .

$\frac{1}{2} \sin^{3} (\frac{π}{2})$

Step 3.4

Simplify the answer.

Tap for more steps...

Step 3.4.1

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$\frac{1}{2} \cdot 1^{3}$

Step 3.4.2

One to any power is one.

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

Step 3.4.3

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

$\frac{1}{2}$

Step 4

Set up the limit as a right-sided limit.

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{1 - \sin (θ)}{\tan^{- 2} (θ)}$

Step 5

Evaluate the right-sided limit.

Tap for more steps...

Step 5.1

Apply L'Hospital's rule.

Tap for more steps...

Step 5.1.1

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

Tap for more steps...

Step 5.1.1.1

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

$\frac{lim_{θ \to {(\frac{π}{2})}^{+}} 1 - \sin (θ)}{lim_{θ \to {(\frac{π}{2})}^{+}} \tan^{- 2} (θ)}$

Step 5.1.1.2

Evaluate the limit of the numerator.

Tap for more steps...

Step 5.1.1.2.1

Evaluate the limit.

Tap for more steps...

Step 5.1.1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as $θ$ approaches $\frac{π}{2}$ .

$\frac{lim_{θ \to {(\frac{π}{2})}^{+}} 1 - lim_{θ \to {(\frac{π}{2})}^{+}} \sin (θ)}{lim_{θ \to {(\frac{π}{2})}^{+}} \tan^{- 2} (θ)}$

Step 5.1.1.2.1.2

Evaluate the limit of $1$ which is constant as $θ$ approaches $\frac{π}{2}$ .

$\frac{1 - lim_{θ \to {(\frac{π}{2})}^{+}} \sin (θ)}{lim_{θ \to {(\frac{π}{2})}^{+}} \tan^{- 2} (θ)}$

Step 5.1.1.2.1.3

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

$\frac{1 - \sin (lim_{θ \to {(\frac{π}{2})}^{+}} θ)}{lim_{θ \to {(\frac{π}{2})}^{+}} \tan^{- 2} (θ)}$

Step 5.1.1.2.2

Evaluate the limit of $θ$ by plugging in $\frac{π}{2}$ for $θ$ .

$\frac{1 - \sin (\frac{π}{2})}{lim_{θ \to {(\frac{π}{2})}^{+}} \tan^{- 2} (θ)}$

Step 5.1.1.2.3

Simplify the answer.

Tap for more steps...

Step 5.1.1.2.3.1

Simplify each term.

Tap for more steps...

Step 5.1.1.2.3.1.1

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$\frac{1 - 1 \cdot 1}{lim_{θ \to {(\frac{π}{2})}^{+}} \tan^{- 2} (θ)}$

Step 5.1.1.2.3.1.2

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{lim_{θ \to {(\frac{π}{2})}^{+}} \tan^{- 2} (θ)}$

Step 5.1.1.2.3.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{θ \to {(\frac{π}{2})}^{+}} \tan^{- 2} (θ)}$

Step 5.1.1.3

Evaluate the limit of the denominator.

Tap for more steps...

Step 5.1.1.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{0}{lim_{θ \to {(\frac{π}{2})}^{+}} \frac{1}{\tan^{2} (θ)}}$

Step 5.1.1.3.2

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{\tan^{2} (θ)}$ approaches $0$ .

$\frac{0}{0}$

Step 5.1.1.3.3

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

Undefined

$\frac{0}{0}$

Step 5.1.1.4

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

Undefined

$\frac{0}{0}$

Step 5.1.2

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.

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{1 - \sin (θ)}{\tan^{- 2} (θ)} = lim_{θ \to {(\frac{π}{2})}^{+}} \frac{\frac{d}{d θ} [1 - \sin (θ)]}{\frac{d}{d θ} [\tan^{- 2} (θ)]}$

Step 5.1.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 5.1.3.1

Differentiate the numerator and denominator.

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

Step 5.1.3.2

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

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

Step 5.1.3.3

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

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

Step 5.1.3.4

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

Tap for more steps...

Step 5.1.3.4.1

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

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

Step 5.1.3.4.2

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

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

Step 5.1.3.5

Subtract $\cos (θ)$ from $0$ .

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{- \cos (θ)}{\frac{d}{d θ} [\tan^{- 2} (θ)]}$

Step 5.1.3.6

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 $g (θ) = \tan (θ)$ .

Tap for more steps...

Step 5.1.3.6.1

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

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{- \cos (θ)}{\frac{d}{d u} [u^{- 2}] \frac{d}{d θ} [\tan (θ)]}$

Step 5.1.3.6.2

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

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{- \cos (θ)}{- 2 u^{- 3} \frac{d}{d θ} [\tan (θ)]}$

Step 5.1.3.6.3

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

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{- \cos (θ)}{- 2 \tan^{- 3} (θ) \frac{d}{d θ} [\tan (θ)]}$

Step 5.1.3.7

The derivative of $\tan (θ)$ with respect to $θ$ is $\sec^{2} (θ)$ .

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{- \cos (θ)}{- 2 \tan^{- 3} (θ) \sec^{2} (θ)}$

Step 5.1.3.8

Simplify.

Tap for more steps...

Step 5.1.3.8.1

Reorder the factors of $- 2 \tan^{- 3} (θ) \sec^{2} (θ)$ .

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{- \cos (θ)}{- 2 \sec^{2} (θ) \tan^{- 3} (θ)}$

Step 5.1.3.8.2

Rewrite $\sec (θ)$ in terms of sines and cosines.

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{- \cos (θ)}{- 2 {(\frac{1}{\cos (θ)})}^{2} \tan^{- 3} (θ)}$

Step 5.1.3.8.3

Apply the product rule to $\frac{1}{\cos (θ)}$ .

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{- \cos (θ)}{- 2 \frac{1^{2}}{\cos^{2} (θ)} \tan^{- 3} (θ)}$

Step 5.1.3.8.4

One to any power is one.

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{- \cos (θ)}{- 2 \frac{1}{\cos^{2} (θ)} \tan^{- 3} (θ)}$

Step 5.1.3.8.5

Combine $- 2$ and $\frac{1}{\cos^{2} (θ)}$ .

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{- \cos (θ)}{\frac{- 2}{\cos^{2} (θ)} \tan^{- 3} (θ)}$

Step 5.1.3.8.6

Move the negative in front of the fraction.

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{- \cos (θ)}{- \frac{2}{\cos^{2} (θ)} \tan^{- 3} (θ)}$

Step 5.1.3.8.7

Rewrite $\tan (θ)$ in terms of sines and cosines.

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{- \cos (θ)}{- \frac{2}{\cos^{2} (θ)} {(\frac{\sin (θ)}{\cos (θ)})}^{- 3}}$

Step 5.1.3.8.8

Change the sign of the exponent by rewriting the base as its reciprocal.

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{- \cos (θ)}{- \frac{2}{\cos^{2} (θ)} {(\frac{\cos (θ)}{\sin (θ)})}^{3}}$

Step 5.1.3.8.9

Apply the product rule to $\frac{\cos (θ)}{\sin (θ)}$ .

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{- \cos (θ)}{- \frac{2}{\cos^{2} (θ)} \cdot \frac{\cos^{3} (θ)}{\sin^{3} (θ)}}$

Step 5.1.3.8.10

Cancel the common factor of $\cos^{2} (θ)$ .

Tap for more steps...

Step 5.1.3.8.10.1

Move the leading negative in $- \frac{2}{\cos^{2} (θ)}$ into the numerator.

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{- \cos (θ)}{\frac{- 2}{\cos^{2} (θ)} \cdot \frac{\cos^{3} (θ)}{\sin^{3} (θ)}}$

Step 5.1.3.8.10.2

Factor $\cos^{2} (θ)$ out of $\cos^{3} (θ)$ .

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{- \cos (θ)}{\frac{- 2}{\cos^{2} (θ)} \cdot \frac{\cos^{2} (θ) \cos (θ)}{\sin^{3} (θ)}}$

Step 5.1.3.8.10.3

Cancel the common factor.

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{- \cos (θ)}{\frac{- 2}{\cos^{2} (θ)} \cdot \frac{\cos^{2} (θ) \cos (θ)}{\sin^{3} (θ)}}$

Step 5.1.3.8.10.4

Rewrite the expression.

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{- \cos (θ)}{- 2 \frac{\cos (θ)}{\sin^{3} (θ)}}$

Step 5.1.3.8.11

Combine $- 2$ and $\frac{\cos (θ)}{\sin^{3} (θ)}$ .

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{- \cos (θ)}{\frac{- 2 \cos (θ)}{\sin^{3} (θ)}}$

Step 5.1.3.8.12

Move the negative in front of the fraction.

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{- \cos (θ)}{- \frac{2 \cos (θ)}{\sin^{3} (θ)}}$

Step 5.1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{θ \to {(\frac{π}{2})}^{+}} - \cos (θ) (- \frac{\sin^{3} (θ)}{2 \cos (θ)})$

Step 5.1.5

Combine factors.

Tap for more steps...

Step 5.1.5.1

Multiply $- 1$ by $- 1$ .

$lim_{θ \to {(\frac{π}{2})}^{+}} 1 \cos (θ) \frac{\sin^{3} (θ)}{2 \cos (θ)}$

Step 5.1.5.2

Multiply $\cos (θ)$ by $1$ .

$lim_{θ \to {(\frac{π}{2})}^{+}} \cos (θ) \frac{\sin^{3} (θ)}{2 \cos (θ)}$

Step 5.1.5.3

Combine $\cos (θ)$ and $\frac{\sin^{3} (θ)}{2 \cos (θ)}$ .

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{\cos (θ) \sin^{3} (θ)}{2 \cos (θ)}$

Step 5.1.6

Cancel the common factor of $\cos (θ)$ .

Tap for more steps...

Step 5.1.6.1

Cancel the common factor.

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{\cos (θ) \sin^{3} (θ)}{2 \cos (θ)}$

Step 5.1.6.2

Rewrite the expression.

$lim_{θ \to {(\frac{π}{2})}^{+}} \frac{\sin^{3} (θ)}{2}$

Step 5.2

Evaluate the limit.

Tap for more steps...

Step 5.2.1

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

$\frac{1}{2} lim_{θ \to {(\frac{π}{2})}^{+}} \sin^{3} (θ)$

Step 5.2.2

Move the exponent $3$ from $\sin^{3} (θ)$ outside the limit using the Limits Power Rule.

$\frac{1}{2} {(lim_{θ \to {(\frac{π}{2})}^{+}} \sin (θ))}^{3}$

Step 5.2.3

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

$\frac{1}{2} \sin^{3} (lim_{θ \to {(\frac{π}{2})}^{+}} θ)$

Step 5.3

Evaluate the limit of $θ$ by plugging in $\frac{π}{2}$ for $θ$ .

$\frac{1}{2} \sin^{3} (\frac{π}{2})$

Step 5.4

Simplify the answer.

Tap for more steps...

Step 5.4.1

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$\frac{1}{2} \cdot 1^{3}$

Step 5.4.2

One to any power is one.

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

Step 5.4.3

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

$\frac{1}{2}$

Step 6

Since the left-sided limit is equal to the right-sided limit, the limit is equal to $\frac{1}{2}$ .

$\frac{1}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$