Calculus Examples

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

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 $0$ .

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

Step 1.2

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

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

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.

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

Step 3

Move the term $\frac{2}{3}$ outside of the limit because it is constant with respect to $x$ .

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

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.

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

Step 4.1.2

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

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

Step 4.1.3

Evaluate the limit of the denominator.

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Step 4.1.3.1

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

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

Step 4.1.3.2

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

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

Step 4.1.3.3

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

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

Step 4.1.3.4

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

Undefined

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

Step 4.1.4

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

Undefined

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

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.

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

Step 4.3

Find the derivative of the numerator and denominator.

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Step 4.3.1

Differentiate the numerator and denominator.

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

Step 4.3.2

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

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

Step 4.3.3

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

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

Step 5

Evaluate the limit.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

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

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

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 $1$ .

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

Step 7.1.2

Convert from $\frac{1}{\cos (0)}$ to $\sec (0)$ .

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

Step 7.1.3

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

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

Step 7.1.4

Multiply $\frac{2}{3}$ by $1$ .

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

Step 7.2

Combine the numerators over the common denominator.

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

Step 7.3

Add $1$ and $2$ .

$\frac{3}{3}$

Step 7.4

Divide $3$ by $3$ .

$1$