Calculus Examples

lim x \to 0 \frac{sin (x)}{3 x} + \frac{2 x}{tan (4 x)}

$lim_{x \to 0} \frac{\sin (x)}{3 x} + \frac{2 x}{\tan (4 x)}$

Step 1

Evaluate the limit.

Tap for more steps...

Step 1.1

Split the limit using the Sum of Limits Rule on the limit as

$x$ approaches

$0$ .

$lim_{x \to 0} \frac{\sin (x)}{3 x} + lim_{x \to 0} \frac{2 x}{\tan (4 x)}$

Step 1.2

Move the term

$\frac{1}{3}$ outside of the limit because it is constant with respect to

$x$ .

$\frac{1}{3} lim_{x \to 0} \frac{\sin (x)}{x} + lim_{x \to 0} \frac{2 x}{\tan (4 x)}$

Step 2

Since

$\cos (x) \leq \frac{\sin (x)}{x} \leq 1$ and

$lim_{x \to 0} \cos (x) = lim_{x \to 0} 1 = 1$ , apply the squeeze theorem.

$\frac{1}{3} \cdot 1 + lim_{x \to 0} \frac{2 x}{\tan (4 x)}$

Step 3

Move the term

$2$ outside of the limit because it is constant with respect to

$x$ .

$\frac{1}{3} \cdot 1 + 2 lim_{x \to 0} \frac{x}{\tan (4 x)}$

Step 4

Apply L'Hospital's rule.

Tap for more steps...

Step 4.1

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

Tap for more steps...

Step 4.1.1

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

$\frac{1}{3} \cdot 1 + 2 \frac{lim_{x \to 0} x}{lim_{x \to 0} \tan (4 x)}$

Step 4.1.2

Evaluate the limit of

$x$ by plugging in

$0$ for

$x$ .

$\frac{1}{3} \cdot 1 + 2 \frac{0}{lim_{x \to 0} \tan (4 x)}$

Step 4.1.3

Evaluate the limit of the denominator.

Tap for more steps...

Step 4.1.3.1

Evaluate the limit.

Tap for more steps...

Step 4.1.3.1.1

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

$\frac{1}{3} \cdot 1 + 2 \frac{0}{\tan (lim_{x \to 0} 4 x)}$

Step 4.1.3.1.2

Move the term

$4$ outside of the limit because it is constant with respect to

$x$ .

$\frac{1}{3} \cdot 1 + 2 \frac{0}{\tan (4 lim_{x \to 0} x)}$

Step 4.1.3.2

Evaluate the limit of

$x$ by plugging in

$0$ for

$x$ .

$\frac{1}{3} \cdot 1 + 2 \frac{0}{\tan (4 \cdot 0)}$

Step 4.1.3.3

Simplify the answer.

Tap for more steps...

Step 4.1.3.3.1

Multiply

$4$ by

$0$ .

$\frac{1}{3} \cdot 1 + 2 \frac{0}{\tan (0)}$

Step 4.1.3.3.2

The exact value of

$\tan (0)$ is

$0$ .

$\frac{1}{3} \cdot 1 + 2 (\frac{0}{0})$

Step 4.1.3.3.3

The expression contains a division by

$0$ . The expression is undefined.

Undefined

$\frac{1}{3} \cdot 1 + 2 (\frac{0}{0})$

Step 4.1.3.4

The expression contains a division by

$0$ . The expression is undefined.

Undefined

$\frac{1}{3} \cdot 1 + 2 (\frac{0}{0})$

Step 4.1.4

The expression contains a division by

$0$ . The expression is undefined.

Undefined

$\frac{1}{3} \cdot 1 + 2 (\frac{0}{0})$

Step 4.2

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.

$lim_{x \to 0} \frac{x}{\tan (4 x)} = lim_{x \to 0} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [\tan (4 x)]}$

Step 4.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 4.3.1

Differentiate the numerator and denominator.

$\frac{1}{3} \cdot 1 + 2 lim_{x \to 0} \frac{\frac{d}{d x} [x]}{\frac{d}{d x} [\tan (4 x)]}$

Step 4.3.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$\frac{1}{3} \cdot 1 + 2 lim_{x \to 0} \frac{1}{\frac{d}{d x} [\tan (4 x)]}$

Step 4.3.3

Differentiate using the chain rule, which states that

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = \tan (x)$ and

$g (x) = 4 x$ .

Tap for more steps...

Step 4.3.3.1

To apply the Chain Rule, set

$u$ as

$4 x$ .

$\frac{1}{3} \cdot 1 + 2 lim_{x \to 0} \frac{1}{\frac{d}{d u} [\tan (u)] \frac{d}{d x} [4 x]}$

Step 4.3.3.2

The derivative of

$\tan (u)$ with respect to

$u$ is

$\sec^{2} (u)$ .

$\frac{1}{3} \cdot 1 + 2 lim_{x \to 0} \frac{1}{\sec^{2} (u) \frac{d}{d x} [4 x]}$

Step 4.3.3.3

Replace all occurrences of

$u$ with

$4 x$ .

$\frac{1}{3} \cdot 1 + 2 lim_{x \to 0} \frac{1}{\sec^{2} (4 x) \frac{d}{d x} [4 x]}$

Step 4.3.4

Since

$4$ is constant with respect to

$x$ , the derivative of

$4 x$ with respect to

$x$ is

$4 \frac{d}{d x} [x]$ .

$\frac{1}{3} \cdot 1 + 2 lim_{x \to 0} \frac{1}{\sec^{2} (4 x) \cdot 4 \frac{d}{d x} [x]}$

Step 4.3.5

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$\frac{1}{3} \cdot 1 + 2 lim_{x \to 0} \frac{1}{\sec^{2} (4 x) \cdot 4 \cdot 1}$

Step 4.3.6

Multiply

$4$ by

$1$ .

$\frac{1}{3} \cdot 1 + 2 lim_{x \to 0} \frac{1}{\sec^{2} (4 x) \cdot 4}$

Step 4.3.7

Move

$4$ to the left of

$\sec^{2} (4 x)$ .

$\frac{1}{3} \cdot 1 + 2 lim_{x \to 0} \frac{1}{4 \sec^{2} (4 x)}$

Step 5

Evaluate the limit.

Tap for more steps...

Step 5.1

Move the term

$\frac{1}{4}$ outside of the limit because it is constant with respect to

$x$ .

$\frac{1}{3} \cdot 1 + 2 (\frac{1}{4}) lim_{x \to 0} \frac{1}{\sec^{2} (4 x)}$

Step 5.2

Split the limit using the Limits Quotient Rule on the limit as

$x$ approaches

$0$ .

$\frac{1}{3} \cdot 1 + 2 (\frac{1}{4}) \frac{lim_{x \to 0} 1}{lim_{x \to 0} \sec^{2} (4 x)}$

Step 5.3

Evaluate the limit of

$1$ which is constant as

$x$ approaches

$0$ .

$\frac{1}{3} \cdot 1 + 2 (\frac{1}{4}) \frac{1}{lim_{x \to 0} \sec^{2} (4 x)}$

Step 5.4

Move the exponent

$2$ from

$\sec^{2} (4 x)$ outside the limit using the Limits Power Rule.

$\frac{1}{3} \cdot 1 + 2 (\frac{1}{4}) \frac{1}{{(lim_{x \to 0} \sec (4 x))}^{2}}$

Step 5.5

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

$\frac{1}{3} \cdot 1 + 2 (\frac{1}{4}) \frac{1}{\sec^{2} (lim_{x \to 0} 4 x)}$

Step 5.6

Move the term

$4$ outside of the limit because it is constant with respect to

$x$ .

$\frac{1}{3} \cdot 1 + 2 (\frac{1}{4}) \frac{1}{\sec^{2} (4 lim_{x \to 0} x)}$

Step 6

Evaluate the limit of

$x$ by plugging in

$0$ for

$x$ .

$\frac{1}{3} \cdot 1 + 2 (\frac{1}{4}) \frac{1}{\sec^{2} (4 \cdot 0)}$

Step 7

Simplify the answer.

Tap for more steps...

Step 7.1

Simplify each term.

Tap for more steps...

Step 7.1.1

Multiply

$\frac{1}{3}$ by

$1$ .

$\frac{1}{3} + 2 (\frac{1}{4}) \frac{1}{\sec^{2} (4 \cdot 0)}$

Step 7.1.2

Cancel the common factor of

$2$ .

Tap for more steps...

Step 7.1.2.1

Factor

$2$ out of

$4$ .

$\frac{1}{3} + 2 \frac{1}{2 (2)} \frac{1}{\sec^{2} (4 \cdot 0)}$

Step 7.1.2.2

Cancel the common factor.

$\frac{1}{3} + 2 \frac{1}{2 \cdot 2} \frac{1}{\sec^{2} (4 \cdot 0)}$

Step 7.1.2.3

Rewrite the expression.

$\frac{1}{3} + \frac{1}{2} \cdot \frac{1}{\sec^{2} (4 \cdot 0)}$

Step 7.1.3

Combine.

$\frac{1}{3} + \frac{1 \cdot 1}{2 \sec^{2} (4 \cdot 0)}$

Step 7.1.4

Multiply

$1$ by

$1$ .

$\frac{1}{3} + \frac{1}{2 \sec^{2} (4 \cdot 0)}$

Step 7.1.5

Simplify the denominator.

Tap for more steps...

Step 7.1.5.1

Multiply

$4$ by

$0$ .

$\frac{1}{3} + \frac{1}{2 \sec^{2} (0)}$

Step 7.1.5.2

The exact value of

$\sec (0)$ is

$1$ .

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

Step 7.1.5.3

One to any power is one.

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

Step 7.1.6

Multiply

$2$ by

$1$ .

$\frac{1}{3} + \frac{1}{2}$

Step 7.2

To write

$\frac{1}{3}$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

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

Step 7.3

To write

$\frac{1}{2}$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

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

Step 7.4

Write each expression with a common denominator of

$6$ , by multiplying each by an appropriate factor of

$1$ .

Tap for more steps...

Step 7.4.1

Multiply

$\frac{1}{3}$ by

$\frac{2}{2}$ .

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

Step 7.4.2

Multiply

$3$ by

$2$ .

$\frac{2}{6} + \frac{1}{2} \cdot \frac{3}{3}$

Step 7.4.3

Multiply

$\frac{1}{2}$ by

$\frac{3}{3}$ .

$\frac{2}{6} + \frac{3}{2 \cdot 3}$

Step 7.4.4

Multiply

$2$ by

$3$ .

$\frac{2}{6} + \frac{3}{6}$

Step 7.5

Combine the numerators over the common denominator.

$\frac{2 + 3}{6}$

Step 7.6

Add

$2$ and

$3$ .

$\frac{5}{6}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{6}$

Decimal Form:

$0.8\overline{3}$