Calculus Examples

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

Step 1

Evaluate the limit.

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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.

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Step 4.1

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

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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.

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Step 4.1.3.1

Evaluate the limit.

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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.

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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.

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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 $g (x) = 4 x$ .

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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.

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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.

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Step 7.1

Simplify each term.

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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 $2$ .

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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.

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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 $1$ .

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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:

$0.8\overline{3}$