Calculus Examples

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

Step 1

Combine terms.

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Step 1.1

To write $\frac{\sin (x)}{x}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

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

Step 1.2

Write each expression with a common denominator of $x^{3}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.1

Multiply $\frac{\sin (x)}{x}$ by $\frac{x^{2}}{x^{2}}$ .

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

Step 1.2.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.2.2.1

Multiply $x$ by $x^{2}$ .

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Step 1.2.2.1.1

Raise $x$ to the power of $1$ .

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

Step 1.2.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.2.2.2

Add $1$ and $2$ .

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

Step 1.3

Combine the numerators over the common denominator.

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

Step 2

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

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

Step 3

Since the numerator is a constant and the denominator approaches $0$ when $x$ approaches $0$ , the fraction $\frac{2}{x^{2}}$ approaches infinity.

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

Step 4

Since $\frac{\tan (2 x) - x^{2}}{x^{3}} \leq \frac{\tan (2 x) + \sin (x) x^{2}}{x^{3}} \leq \frac{\tan (2 x) + 1 x^{2}}{x^{3}}$ and $lim_{x \to 0} \frac{\tan (2 x) - x^{2}}{x^{3}} = lim_{x \to 0} \frac{\tan (2 x) + 1 x^{2}}{x^{3}} = \infty$ , apply the squeeze theorem.

$\infty + \infty$

Step 5

Infinity plus infinity is infinity.

$\infty$