Calculus Examples

$lim_{x \to 0} \frac{\sin (3 x)}{x + \tan (4 x)}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

$\frac{lim_{x \to 0} \sin (3 x)}{lim_{x \to 0} x + \tan (4 x)}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

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

$\frac{\sin (lim_{x \to 0} 3 x)}{lim_{x \to 0} x + \tan (4 x)}$

Step 1.1.2.1.2

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sin (3 lim_{x \to 0} x)}{lim_{x \to 0} x + \tan (4 x)}$

Step 1.1.2.2

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

$\frac{\sin (3 \cdot 0)}{lim_{x \to 0} x + \tan (4 x)}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Multiply $3$ by $0$ .

$\frac{\sin (0)}{lim_{x \to 0} x + \tan (4 x)}$

Step 1.1.2.3.2

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

$\frac{0}{lim_{x \to 0} x + \tan (4 x)}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

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

$\frac{0}{lim_{x \to 0} x + lim_{x \to 0} \tan (4 x)}$

Step 1.1.3.2

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

$\frac{0}{lim_{x \to 0} x + \tan (lim_{x \to 0} 4 x)}$

Step 1.1.3.3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{0}{lim_{x \to 0} x + \tan (4 lim_{x \to 0} x)}$

Step 1.1.3.4

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

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Step 1.1.3.4.1

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

$\frac{0}{0 + \tan (4 lim_{x \to 0} x)}$

Step 1.1.3.4.2

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

$\frac{0}{0 + \tan (4 \cdot 0)}$

Step 1.1.3.5

Simplify the answer.

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Step 1.1.3.5.1

Simplify each term.

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Step 1.1.3.5.1.1

Multiply $4$ by $0$ .

$\frac{0}{0 + \tan (0)}$

Step 1.1.3.5.1.2

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

$\frac{0}{0 + 0}$

Step 1.1.3.5.2

Add $0$ and $0$ .

$\frac{0}{0}$

Step 1.1.3.5.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.6

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

Undefined

$\frac{0}{0}$

Step 1.1.4

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

Undefined

$\frac{0}{0}$

Step 1.2

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.

$lim_{x \to 0} \frac{\sin (3 x)}{x + \tan (4 x)} = lim_{x \to 0} \frac{\frac{d}{d x} [\sin (3 x)]}{\frac{d}{d x} [x + \tan (4 x)]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [\sin (3 x)]}{\frac{d}{d x} [x + \tan (4 x)]}$

Step 1.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 3 x$ .

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Step 1.3.2.1

To apply the Chain Rule, set $u$ as $3 x$ .

$lim_{x \to 0} \frac{\frac{d}{d u} [\sin (u)] \frac{d}{d x} [3 x]}{\frac{d}{d x} [x + \tan (4 x)]}$

Step 1.3.2.2

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

$lim_{x \to 0} \frac{\cos (u) \frac{d}{d x} [3 x]}{\frac{d}{d x} [x + \tan (4 x)]}$

Step 1.3.2.3

Replace all occurrences of $u$ with $3 x$ .

$lim_{x \to 0} \frac{\cos (3 x) \frac{d}{d x} [3 x]}{\frac{d}{d x} [x + \tan (4 x)]}$

Step 1.3.3

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

$lim_{x \to 0} \frac{\cos (3 x) (3 \frac{d}{d x} [x])}{\frac{d}{d x} [x + \tan (4 x)]}$

Step 1.3.4

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

$lim_{x \to 0} \frac{\cos (3 x) (3 \cdot 1)}{\frac{d}{d x} [x + \tan (4 x)]}$

Step 1.3.5

Multiply $3$ by $1$ .

$lim_{x \to 0} \frac{\cos (3 x) \cdot 3}{\frac{d}{d x} [x + \tan (4 x)]}$

Step 1.3.6

Move $3$ to the left of $\cos (3 x)$ .

$lim_{x \to 0} \frac{3 \cdot \cos (3 x)}{\frac{d}{d x} [x + \tan (4 x)]}$

Step 1.3.7

By the Sum Rule, the derivative of $x + \tan (4 x)$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\tan (4 x)]$ .

$lim_{x \to 0} \frac{3 \cos (3 x)}{\frac{d}{d x} [x] + \frac{d}{d x} [\tan (4 x)]}$

Step 1.3.8

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

$lim_{x \to 0} \frac{3 \cos (3 x)}{1 + \frac{d}{d x} [\tan (4 x)]}$

Step 1.3.9

Evaluate $\frac{d}{d x} [\tan (4 x)]$ .

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Step 1.3.9.1

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 $g (x) = 4 x$ .

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Step 1.3.9.1.1

To apply the Chain Rule, set $u$ as $4 x$ .

$lim_{x \to 0} \frac{3 \cos (3 x)}{1 + \frac{d}{d u} [\tan (u)] \frac{d}{d x} [4 x]}$

Step 1.3.9.1.2

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

$lim_{x \to 0} \frac{3 \cos (3 x)}{1 + \sec^{2} (u) \frac{d}{d x} [4 x]}$

Step 1.3.9.1.3

Replace all occurrences of $u$ with $4 x$ .

$lim_{x \to 0} \frac{3 \cos (3 x)}{1 + \sec^{2} (4 x) \frac{d}{d x} [4 x]}$

Step 1.3.9.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$lim_{x \to 0} \frac{3 \cos (3 x)}{1 + \sec^{2} (4 x) (4 \frac{d}{d x} [x])}$

Step 1.3.9.3

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

$lim_{x \to 0} \frac{3 \cos (3 x)}{1 + \sec^{2} (4 x) (4 \cdot 1)}$

Step 1.3.9.4

Multiply $4$ by $1$ .

$lim_{x \to 0} \frac{3 \cos (3 x)}{1 + \sec^{2} (4 x) \cdot 4}$

Step 1.3.9.5

Move $4$ to the left of $\sec^{2} (4 x)$ .

$lim_{x \to 0} \frac{3 \cos (3 x)}{1 + 4 \sec^{2} (4 x)}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$3 lim_{x \to 0} \frac{\cos (3 x)}{1 + 4 \sec^{2} (4 x)}$

Step 2.2

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

$3 \frac{lim_{x \to 0} \cos (3 x)}{lim_{x \to 0} 1 + 4 \sec^{2} (4 x)}$

Step 2.3

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

$3 \frac{\cos (lim_{x \to 0} 3 x)}{lim_{x \to 0} 1 + 4 \sec^{2} (4 x)}$

Step 2.4

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$3 \frac{\cos (3 lim_{x \to 0} x)}{lim_{x \to 0} 1 + 4 \sec^{2} (4 x)}$

Step 2.5

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

$3 \frac{\cos (3 lim_{x \to 0} x)}{lim_{x \to 0} 1 + lim_{x \to 0} 4 \sec^{2} (4 x)}$

Step 2.6

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

$3 \frac{\cos (3 lim_{x \to 0} x)}{1 + lim_{x \to 0} 4 \sec^{2} (4 x)}$

Step 2.7

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$3 \frac{\cos (3 lim_{x \to 0} x)}{1 + 4 lim_{x \to 0} \sec^{2} (4 x)}$

Step 2.8

Move the exponent $2$ from $\sec^{2} (4 x)$ outside the limit using the Limits Power Rule.

$3 \frac{\cos (3 lim_{x \to 0} x)}{1 + 4 {(lim_{x \to 0} \sec (4 x))}^{2}}$

Step 2.9

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

$3 \frac{\cos (3 lim_{x \to 0} x)}{1 + 4 \sec^{2} (lim_{x \to 0} 4 x)}$

Step 2.10

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$3 \frac{\cos (3 lim_{x \to 0} x)}{1 + 4 \sec^{2} (4 lim_{x \to 0} x)}$

Step 3

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

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Step 3.1

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

$3 \frac{\cos (3 \cdot 0)}{1 + 4 \sec^{2} (4 lim_{x \to 0} x)}$

Step 3.2

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

$3 \frac{\cos (3 \cdot 0)}{1 + 4 \sec^{2} (4 \cdot 0)}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $3$ by $0$ .

$3 \frac{\cos (0)}{1 + 4 \sec^{2} (4 \cdot 0)}$

Step 4.1.2

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

$3 \frac{1}{1 + 4 \sec^{2} (4 \cdot 0)}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply $4$ by $0$ .

$3 \frac{1}{1 + 4 \sec^{2} (0)}$

Step 4.2.2

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

$3 \frac{1}{1 + 4 \cdot 1^{2}}$

Step 4.2.3

One to any power is one.

$3 \frac{1}{1 + 4 \cdot 1}$

Step 4.2.4

Multiply $4$ by $1$ .

$3 \frac{1}{1 + 4}$

Step 4.2.5

Add $1$ and $4$ .

$3 (\frac{1}{5})$

Step 4.3

Combine $3$ and $\frac{1}{5}$ .

$\frac{3}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{5}$

Decimal Form:

$0.6$