Calculus Examples

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

Step 1

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

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

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Evaluate the limit.

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Step 2.1.2.1.1

Move the exponent $2$ from $\tan^{2} (x)$ outside the limit using the Limits Power Rule.

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

Step 2.1.2.1.2

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

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

Step 2.1.2.2

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

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

Step 2.1.2.3

Simplify the answer.

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Step 2.1.2.3.1

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

$2 \frac{0^{2}}{lim_{x \to 0} x}$

Step 2.1.2.3.2

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

$2 \frac{0}{lim_{x \to 0} x}$

Step 2.1.3

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

$2 (\frac{0}{0})$

Step 2.1.4

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

Undefined

$2 (\frac{0}{0})$

Step 2.2

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.

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

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

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

Step 2.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \tan (x)$ .

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Step 2.3.2.1

To apply the Chain Rule, set $u$ as $\tan (x)$ .

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

Step 2.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

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

Step 2.3.2.3

Replace all occurrences of $u$ with $\tan (x)$ .

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

Step 2.3.3

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

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

Step 2.3.4

Reorder the factors of $2 \tan (x) \sec^{2} (x)$ .

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

Step 2.3.5

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

$2 lim_{x \to 0} \frac{2 \sec^{2} (x) \tan (x)}{1}$

Step 2.4

Divide $2 \sec^{2} (x) \tan (x)$ by $1$ .

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

Step 3

Evaluate the limit.

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Step 3.1

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

$2 \cdot 2 lim_{x \to 0} \sec^{2} (x) \tan (x)$

Step 3.2

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

$2 \cdot 2 lim_{x \to 0} \sec^{2} (x) \cdot lim_{x \to 0} \tan (x)$

Step 3.3

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

$2 \cdot 2 {(lim_{x \to 0} \sec (x))}^{2} \cdot lim_{x \to 0} \tan (x)$

Step 3.4

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

$2 \cdot 2 \sec^{2} (lim_{x \to 0} x) \cdot lim_{x \to 0} \tan (x)$

Step 3.5

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

$2 \cdot 2 \sec^{2} (lim_{x \to 0} x) \cdot \tan (lim_{x \to 0} x)$

Step 4

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 4.1

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

$2 \cdot 2 \sec^{2} (0) \cdot \tan (lim_{x \to 0} x)$

Step 4.2

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

$2 \cdot 2 \sec^{2} (0) \cdot \tan (0)$

Step 5

Simplify the answer.

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Step 5.1

Multiply $2$ by $2$ .

$4 \sec^{2} (0) \cdot \tan (0)$

Step 5.2

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

$4 \cdot 1^{2} \cdot \tan (0)$

Step 5.3

One to any power is one.

$4 \cdot 1 \cdot \tan (0)$

Step 5.4

Multiply $4$ by $1$ .

$4 \cdot \tan (0)$

Step 5.5

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

$4 \cdot 0$

Step 5.6

Multiply $4$ by $0$ .

$0$