Calculus Examples

$lim_{θ \to 0} \frac{\sin (θ)}{θ}$

Step 1

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

Tap for more steps...

Step 1.1

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

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

Step 1.2

Evaluate the limit of the numerator.

Tap for more steps...

Step 1.2.1

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

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

Step 1.2.2

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

$\frac{\sin (0)}{lim_{θ \to 0} θ}$

Step 1.2.3

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

$\frac{0}{lim_{θ \to 0} θ}$

Step 1.3

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

$\frac{0}{0}$

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.

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

Step 3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

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

$lim_{θ \to 0} \frac{\cos (θ)}{1}$

Step 4

Evaluate the limit.

Tap for more steps...

Step 4.1

Divide $\cos (θ)$ by $1$ .

$lim_{θ \to 0} \cos (θ)$

Step 4.2

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

$\cos (lim_{θ \to 0} θ)$

Step 5

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

$\cos (0)$

Step 6

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

$1$