Calculus Examples

$lim_{x \to 0} \frac{2 \tan (x)}{5 x^{3}}$

Step 1

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

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Step 1.1

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

$\frac{lim_{x \to 0} 2 \tan (x)}{lim_{x \to 0} 5 x^{3}}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Evaluate the limit.

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Step 1.2.1.1

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

$\frac{2 lim_{x \to 0} \tan (x)}{lim_{x \to 0} 5 x^{3}}$

Step 1.2.1.2

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

$\frac{2 \tan (lim_{x \to 0} x)}{lim_{x \to 0} 5 x^{3}}$

Step 1.2.2

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

$\frac{2 \tan (0)}{lim_{x \to 0} 5 x^{3}}$

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

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

$\frac{2 \cdot 0}{lim_{x \to 0} 5 x^{3}}$

Step 1.2.3.2

Multiply $2$ by $0$ .

$\frac{0}{lim_{x \to 0} 5 x^{3}}$

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Evaluate the limit.

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Step 1.3.1.1

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{5 lim_{x \to 0} x^{3}}$

Step 1.3.1.2

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{0}{5 {(lim_{x \to 0} x)}^{3}}$

Step 1.3.2

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

$\frac{0}{5 \cdot 0^{3}}$

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Raising $0$ to any positive power yields $0$ .

$\frac{0}{5 \cdot 0}$

Step 1.3.3.2

Multiply $5$ by $0$ .

$\frac{0}{0}$

Step 1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.4

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

Undefined

$\frac{0}{0}$

Step 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.

$lim_{x \to 0} \frac{2 \tan (x)}{5 x^{3}} = lim_{x \to 0} \frac{\frac{d}{d x} [2 \tan (x)]}{\frac{d}{d x} [5 x^{3}]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [2 \tan (x)]}{\frac{d}{d x} [5 x^{3}]}$

Step 3.2

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

$lim_{x \to 0} \frac{2 \frac{d}{d x} [\tan (x)]}{\frac{d}{d x} [5 x^{3}]}$

Step 3.3

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$lim_{x \to 0} \frac{2 \sec^{2} (x)}{\frac{d}{d x} [5 x^{3}]}$

Step 3.4

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{3}$ with respect to $x$ is $5 \frac{d}{d x} [x^{3}]$ .

$lim_{x \to 0} \frac{2 \sec^{2} (x)}{5 \frac{d}{d x} [x^{3}]}$

Step 3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$lim_{x \to 0} \frac{2 \sec^{2} (x)}{5 (3 x^{2})}$

Step 3.6

Multiply $3$ by $5$ .

$lim_{x \to 0} \frac{2 \sec^{2} (x)}{15 x^{2}}$

Step 4

Since the numerator is positive and the denominator $15 x^{2}$ approaches zero and is greater than zero for $x$ near $0$ on both sides, the function increases without bound.

$\infty$