Calculus Examples

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

Step 1

Multiply the numerator and denominator by $x$ .

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

Step 2

Multiply the numerator and denominator by $2 x$ .

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

Step 3

Separate fractions.

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

Step 4

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

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

Step 5

The limit of $\frac{\sin (2 x)}{2 x}$ as $x$ approaches $0$ is $1$ .

Tap for more steps...

Step 5.1

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

Tap for more steps...

Step 5.1.1

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

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

Step 5.1.2

Evaluate the limit of the numerator.

Tap for more steps...

Step 5.1.2.1

Evaluate the limit.

Tap for more steps...

Step 5.1.2.1.1

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

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

Step 5.1.2.1.2

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

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

Step 5.1.2.2

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

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

Step 5.1.2.3

Simplify the answer.

Tap for more steps...

Step 5.1.2.3.1

Multiply $2$ by $0$ .

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

Step 5.1.2.3.2

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

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

Step 5.1.3

Evaluate the limit of the denominator.

Tap for more steps...

Step 5.1.3.1

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

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

Step 5.1.3.2

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

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

Step 5.1.3.3

Multiply $2$ by $0$ .

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

Step 5.1.3.4

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

Undefined

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

Step 5.1.4

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

Undefined

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

Step 5.2

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.

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

Step 5.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 5.3.1

Differentiate the numerator and denominator.

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

Step 5.3.2

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

Tap for more steps...

Step 5.3.2.1

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

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

Step 5.3.2.2

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

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

Step 5.3.2.3

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

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

Step 5.3.3

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

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

Step 5.3.4

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

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

Step 5.3.5

Multiply $2$ by $1$ .

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

Step 5.3.6

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

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

Step 5.3.7

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

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

Step 5.3.8

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

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

Step 5.3.9

Multiply $2$ by $1$ .

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

Step 5.4

Evaluate the limit.

Tap for more steps...

Step 5.4.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.4.1.1

Cancel the common factor.

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

Step 5.4.1.2

Divide $\cos (2 x)$ by $1$ .

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

Step 5.4.2

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

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

Step 5.4.3

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

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

Step 5.5

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

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

Step 5.6

Simplify the answer.

Tap for more steps...

Step 5.6.1

Multiply $2$ by $0$ .

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

Step 5.6.2

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

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

Step 6

The limit of $\frac{x}{\sin (x)}$ as $x$ approaches $0$ is $1$ .

Tap for more steps...

Step 6.1

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

Tap for more steps...

Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.1.3

Evaluate the limit of the denominator.

Tap for more steps...

Step 6.1.3.1

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

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

Step 6.1.3.2

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

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

Step 6.1.3.3

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

$1 \cdot \frac{0}{0} \cdot lim_{x \to 0} \frac{2 x}{x}$

Step 6.1.3.4

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

Undefined

$1 \cdot \frac{0}{0} \cdot lim_{x \to 0} \frac{2 x}{x}$

Step 6.1.4

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

Undefined

$1 \cdot \frac{0}{0} \cdot lim_{x \to 0} \frac{2 x}{x}$

Step 6.2

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 6.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 6.3.1

Differentiate the numerator and denominator.

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

Step 6.3.2

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

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

Step 6.3.3

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

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

Step 6.4

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

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

Step 6.5

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

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

Step 6.6

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

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

Step 6.7

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

$1 \cdot 1 \cdot lim_{x \to 0} \frac{2 x}{x}$

Step 7

Cancel the common factor of $x$ .

Tap for more steps...

Step 7.1

Cancel the common factor.

$1 \cdot 1 \cdot lim_{x \to 0} \frac{2 x}{x}$

Step 7.2

Divide $2$ by $1$ .

$1 \cdot 1 \cdot lim_{x \to 0} 2$

Step 8

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

$1 \cdot 1 \cdot 2$

Step 9

Multiply $1 \cdot 1 \cdot 2$ .

Tap for more steps...

Step 9.1

Multiply $1$ by $1$ .

$1 \cdot 2$

Step 9.2

Multiply $2$ by $1$ .

$2$