Calculus Examples

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

Step 1

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

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

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.

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

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 limit inside the trig function because tangent is continuous.

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

Step 2.1.2.1.2

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

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

Step 2.1.2.2

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

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

Step 2.1.2.3

Simplify the answer.

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Step 2.1.2.3.1

Multiply $2$ by $0$ .

$\frac{1}{3} \cdot \frac{\tan (0)}{lim_{x \to 0} x}$

Step 2.1.2.3.2

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

$\frac{1}{3} \cdot \frac{0}{lim_{x \to 0} x}$

Step 2.1.3

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

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

Step 2.1.4

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

Undefined

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

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.

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

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 $f (x) = \tan (x)$ and $g (x) = 2 x$ .

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Step 2.3.2.1

To apply the Chain Rule, set $u$ as $2 x$ .

$\frac{1}{3} lim_{x \to 0} \frac{\frac{d}{d u} [\tan (u)] \frac{d}{d x} [2 x]}{\frac{d}{d x} [x]}$

Step 2.3.2.2

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

$\frac{1}{3} lim_{x \to 0} \frac{\sec^{2} (u) \frac{d}{d x} [2 x]}{\frac{d}{d x} [x]}$

Step 2.3.2.3

Replace all occurrences of $u$ with $2 x$ .

$\frac{1}{3} lim_{x \to 0} \frac{\sec^{2} (2 x) \frac{d}{d x} [2 x]}{\frac{d}{d x} [x]}$

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Multiply $2$ by $1$ .

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

Step 2.3.6

Move $2$ to the left of $\sec^{2} (2 x)$ .

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

Step 2.3.7

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

$\frac{1}{3} lim_{x \to 0} \frac{2 \sec^{2} (2 x)}{1}$

Step 2.4

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

$\frac{1}{3} lim_{x \to 0} 2 \sec^{2} (2 x)$

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 $x$ .

$\frac{1}{3} \cdot 2 lim_{x \to 0} \sec^{2} (2 x)$

Step 3.2

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

$\frac{1}{3} \cdot 2 {(lim_{x \to 0} \sec (2 x))}^{2}$

Step 3.3

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

$\frac{1}{3} \cdot 2 \sec^{2} (lim_{x \to 0} 2 x)$

Step 3.4

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

$\frac{1}{3} \cdot 2 \sec^{2} (2 lim_{x \to 0} x)$

Step 4

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

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

Step 5

Simplify the answer.

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Step 5.1

Combine $\frac{1}{3}$ and $2$ .

$\frac{2}{3} \sec^{2} (2 \cdot 0)$

Step 5.2

Multiply $2$ by $0$ .

$\frac{2}{3} \sec^{2} (0)$

Step 5.3

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

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

Step 5.4

One to any power is one.

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

Step 5.5

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

$\frac{2}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3}$

Decimal Form:

$0.\overline{6}$