Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite ${(1 + 4 x)}^{\frac{3}{x}}$ as $e^{\ln ({(1 + 4 x)}^{\frac{3}{x}})}$ .

$lim_{x \to 0} e^{\ln ({(1 + 4 x)}^{\frac{3}{x}})}$

Step 1.2

Expand $\ln ({(1 + 4 x)}^{\frac{3}{x}})$ by moving $\frac{3}{x}$ outside the logarithm.

$lim_{x \to 0} e^{\frac{3}{x} \ln (1 + 4 x)}$

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

$e^{lim_{x \to 0} \frac{3}{x} \ln (1 + 4 x)}$

Step 2.2

Combine $\frac{3}{x}$ and $\ln (1 + 4 x)$ .

$e^{lim_{x \to 0} \frac{3 \ln (1 + 4 x)}{x}}$

Step 2.3

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

$e^{3 lim_{x \to 0} \frac{\ln (1 + 4 x)}{x}}$

Step 3

Apply L'Hospital's rule.

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Step 3.1

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

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Step 3.1.1

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

$e^{3 \frac{lim_{x \to 0} \ln (1 + 4 x)}{lim_{x \to 0} x}}$

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

Evaluate the limit.

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Step 3.1.2.1.1

Move the limit inside the logarithm.

$e^{3 \frac{\ln (lim_{x \to 0} 1 + 4 x)}{lim_{x \to 0} x}}$

Step 3.1.2.1.2

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

$e^{3 \frac{\ln (lim_{x \to 0} 1 + lim_{x \to 0} 4 x)}{lim_{x \to 0} x}}$

Step 3.1.2.1.3

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

$e^{3 \frac{\ln (1 + lim_{x \to 0} 4 x)}{lim_{x \to 0} x}}$

Step 3.1.2.1.4

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

$e^{3 \frac{\ln (1 + 4 lim_{x \to 0} x)}{lim_{x \to 0} x}}$

Step 3.1.2.2

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

$e^{3 \frac{\ln (1 + 4 \cdot 0)}{lim_{x \to 0} x}}$

Step 3.1.2.3

Simplify the answer.

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Step 3.1.2.3.1

Multiply $4$ by $0$ .

$e^{3 \frac{\ln (1 + 0)}{lim_{x \to 0} x}}$

Step 3.1.2.3.2

Add $1$ and $0$ .

$e^{3 \frac{\ln (1)}{lim_{x \to 0} x}}$

Step 3.1.2.3.3

The natural logarithm of $1$ is $0$ .

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

Step 3.1.3

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

$e^{3 (\frac{0}{0})}$

Step 3.1.4

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

Undefined

$e^{3 (\frac{0}{0})}$

Step 3.2

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.

$lim_{x \to 0} \frac{\ln (1 + 4 x)}{x} = lim_{x \to 0} \frac{\frac{d}{d x} [\ln (1 + 4 x)]}{\frac{d}{d x} [x]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$e^{3 lim_{x \to 0} \frac{\frac{d}{d x} [\ln (1 + 4 x)]}{\frac{d}{d x} [x]}}$

Step 3.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 1 + 4 x$ .

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Step 3.3.2.1

To apply the Chain Rule, set $u$ as $1 + 4 x$ .

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

Step 3.3.2.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$e^{3 lim_{x \to 0} \frac{\frac{1}{u} \frac{d}{d x} [1 + 4 x]}{\frac{d}{d x} [x]}}$

Step 3.3.2.3

Replace all occurrences of $u$ with $1 + 4 x$ .

$e^{3 lim_{x \to 0} \frac{\frac{1}{1 + 4 x} \frac{d}{d x} [1 + 4 x]}{\frac{d}{d x} [x]}}$

Step 3.3.3

By the Sum Rule, the derivative of $1 + 4 x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [4 x]$ .

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

Step 3.3.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 3.3.5

Add $0$ and $\frac{d}{d x} [4 x]$ .

$e^{3 lim_{x \to 0} \frac{\frac{1}{1 + 4 x} \frac{d}{d x} [4 x]}{\frac{d}{d x} [x]}}$

Step 3.3.6

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

$e^{3 lim_{x \to 0} \frac{\frac{1}{1 + 4 x} \cdot 4 \frac{d}{d x} [x]}{\frac{d}{d x} [x]}}$

Step 3.3.7

Combine $4$ and $\frac{1}{1 + 4 x}$ .

$e^{3 lim_{x \to 0} \frac{\frac{4}{1 + 4 x} \frac{d}{d x} [x]}{\frac{d}{d x} [x]}}$

Step 3.3.8

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

$e^{3 lim_{x \to 0} \frac{\frac{4}{1 + 4 x} \cdot 1}{\frac{d}{d x} [x]}}$

Step 3.3.9

Multiply $\frac{4}{1 + 4 x}$ by $1$ .

$e^{3 lim_{x \to 0} \frac{\frac{4}{1 + 4 x}}{\frac{d}{d x} [x]}}$

Step 3.3.10

Reorder terms.

$e^{3 lim_{x \to 0} \frac{\frac{4}{4 x + 1}}{\frac{d}{d x} [x]}}$

Step 3.3.11

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

$e^{3 lim_{x \to 0} \frac{\frac{4}{4 x + 1}}{1}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$e^{3 lim_{x \to 0} \frac{4}{4 x + 1} \cdot 1}$

Step 3.5

Multiply $\frac{4}{4 x + 1}$ by $1$ .

$e^{3 lim_{x \to 0} \frac{4}{4 x + 1}}$

Step 4

Evaluate the limit.

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Step 4.1

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

$e^{3 \cdot 4 lim_{x \to 0} \frac{1}{4 x + 1}}$

Step 4.2

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

$e^{3 \cdot 4 \frac{lim_{x \to 0} 1}{lim_{x \to 0} 4 x + 1}}$

Step 4.3

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

$e^{3 \cdot 4 \frac{1}{lim_{x \to 0} 4 x + 1}}$

Step 4.4

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

$e^{3 \cdot 4 \frac{1}{lim_{x \to 0} 4 x + lim_{x \to 0} 1}}$

Step 4.5

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

$e^{3 \cdot 4 \frac{1}{4 lim_{x \to 0} x + lim_{x \to 0} 1}}$

Step 4.6

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

$e^{3 \cdot 4 \frac{1}{4 lim_{x \to 0} x + 1}}$

Step 5

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

$e^{3 \cdot 4 \frac{1}{4 \cdot 0 + 1}}$

Step 6

Simplify the answer.

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Step 6.1

Multiply $3$ by $4$ .

$e^{12 \frac{1}{4 \cdot 0 + 1}}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Multiply $4$ by $0$ .

$e^{12 \frac{1}{0 + 1}}$

Step 6.2.2

Add $0$ and $1$ .

$e^{12 (\frac{1}{1})}$

Step 6.3

Cancel the common factor of $1$ .

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Step 6.3.1

Cancel the common factor.

$e^{12 (\frac{1}{1})}$

Step 6.3.2

Rewrite the expression.

$e^{12 \cdot 1}$

Step 6.4

Multiply $12$ by $1$ .

$e^{12}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$e^{12}$

Decimal Form:

$162754.79141900 \dots$