Calculus Examples

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

Step 1

Apply L'Hospital's rule.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

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

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

Step 1.1.2

Evaluate the limit of the numerator.

Tap for more steps...

Step 1.1.2.1

Evaluate the limit.

Tap for more steps...

Step 1.1.2.1.1

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

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

Step 1.1.2.1.2

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

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

Step 1.1.2.1.3

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Simplify the answer.

Tap for more steps...

Step 1.1.2.3.1

Simplify each term.

Tap for more steps...

Step 1.1.2.3.1.1

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

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

Step 1.1.2.3.1.2

Multiply $- 1$ by $1$ .

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

Step 1.1.2.3.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{θ \to 0} \sin (θ)}$

Step 1.1.3

Evaluate the limit of the denominator.

Tap for more steps...

Step 1.1.3.1

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

$\frac{0}{\sin (lim_{θ \to 0} θ)}$

Step 1.1.3.2

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

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

Step 1.1.3.3

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

$\frac{0}{0}$

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

Add $- \sin (θ)$ and $0$ .

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

Step 1.3.6

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

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

Step 2

Evaluate the limit.

Tap for more steps...

Step 2.1

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

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

Step 2.2

Move the term $- 1$ outside of the limit because it is constant with respect to $θ$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

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

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 4

Simplify the answer.

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

$\frac{- 0}{1}$

Step 4.3

Multiply $- 1$ by $0$ .

$\frac{0}{1}$

Step 4.4

Divide $0$ by $1$ .

$0$