Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite ${(1 + 2 x)}^{\frac{1}{3 x}}$ as $e^{\ln ({(1 + 2 x)}^{\frac{1}{3 x}})}$ .

$lim_{x \to 0} e^{\ln ({(1 + 2 x)}^{\frac{1}{3 x}})}$

Step 1.2

Expand $\ln ({(1 + 2 x)}^{\frac{1}{3 x}})$ by moving $\frac{1}{3 x}$ outside the logarithm.

$lim_{x \to 0} e^{\frac{1}{3 x} \ln (1 + 2 x)}$

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

$e^{lim_{x \to 0} \frac{1}{3 x} \ln (1 + 2 x)}$

Step 2.2

Combine $\frac{1}{3 x}$ and $\ln (1 + 2 x)$ .

$e^{lim_{x \to 0} \frac{\ln (1 + 2 x)}{3 x}}$

Step 2.3

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

$e^{\frac{1}{3} lim_{x \to 0} \frac{\ln (1 + 2 x)}{x}}$

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.

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

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.

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

Step 3.1.2.1.2

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

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

Step 3.1.2.1.3

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

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

Step 3.1.2.1.4

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

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

Step 3.1.2.2

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

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

Step 3.1.2.3

Simplify the answer.

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Step 3.1.2.3.1

Multiply $2$ by $0$ .

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

Step 3.1.2.3.2

Add $1$ and $0$ .

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

Step 3.1.2.3.3

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

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

Step 3.1.3

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

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

Step 3.1.4

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

Undefined

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

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.

$lim_{x \to 0} \frac{\ln (1 + 2 x)}{x} = lim_{x \to 0} \frac{\frac{d}{d x} [\ln (1 + 2 x)]}{\frac{d}{d x} [x]}$

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

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 $g (x) = 1 + 2 x$ .

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Step 3.3.2.1

To apply the Chain Rule, set $u$ as $1 + 2 x$ .

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

Step 3.3.2.2

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

$e^{\frac{1}{3} lim_{x \to 0} \frac{\frac{1}{u} \frac{d}{d x} [1 + 2 x]}{\frac{d}{d x} [x]}}$

Step 3.3.2.3

Replace all occurrences of $u$ with $1 + 2 x$ .

$e^{\frac{1}{3} lim_{x \to 0} \frac{\frac{1}{1 + 2 x} \frac{d}{d x} [1 + 2 x]}{\frac{d}{d x} [x]}}$

Step 3.3.3

By the Sum Rule, the derivative of $1 + 2 x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [2 x]$ .

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

Step 3.3.4

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

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

Step 3.3.5

Add $0$ and $\frac{d}{d x} [2 x]$ .

$e^{\frac{1}{3} lim_{x \to 0} \frac{\frac{1}{1 + 2 x} \frac{d}{d x} [2 x]}{\frac{d}{d x} [x]}}$

Step 3.3.6

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

$e^{\frac{1}{3} lim_{x \to 0} \frac{\frac{1}{1 + 2 x} \cdot 2 \frac{d}{d x} [x]}{\frac{d}{d x} [x]}}$

Step 3.3.7

Combine $2$ and $\frac{1}{1 + 2 x}$ .

$e^{\frac{1}{3} lim_{x \to 0} \frac{\frac{2}{1 + 2 x} \frac{d}{d x} [x]}{\frac{d}{d x} [x]}}$

Step 3.3.8

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

$e^{\frac{1}{3} lim_{x \to 0} \frac{\frac{2}{1 + 2 x} \cdot 1}{\frac{d}{d x} [x]}}$

Step 3.3.9

Multiply $\frac{2}{1 + 2 x}$ by $1$ .

$e^{\frac{1}{3} lim_{x \to 0} \frac{\frac{2}{1 + 2 x}}{\frac{d}{d x} [x]}}$

Step 3.3.10

Reorder terms.

$e^{\frac{1}{3} lim_{x \to 0} \frac{\frac{2}{2 x + 1}}{\frac{d}{d x} [x]}}$

Step 3.3.11

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

$e^{\frac{1}{3} lim_{x \to 0} \frac{\frac{2}{2 x + 1}}{1}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$e^{\frac{1}{3} lim_{x \to 0} \frac{2}{2 x + 1} \cdot 1}$

Step 3.5

Multiply $\frac{2}{2 x + 1}$ by $1$ .

$e^{\frac{1}{3} lim_{x \to 0} \frac{2}{2 x + 1}}$

Step 4

Evaluate the limit.

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Step 4.1

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

$e^{\frac{1}{3} \cdot 2 lim_{x \to 0} \frac{1}{2 x + 1}}$

Step 4.2

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

$e^{\frac{1}{3} \cdot 2 \frac{lim_{x \to 0} 1}{lim_{x \to 0} 2 x + 1}}$

Step 4.3

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

$e^{\frac{1}{3} \cdot 2 \frac{1}{lim_{x \to 0} 2 x + 1}}$

Step 4.4

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

$e^{\frac{1}{3} \cdot 2 \frac{1}{lim_{x \to 0} 2 x + lim_{x \to 0} 1}}$

Step 4.5

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

$e^{\frac{1}{3} \cdot 2 \frac{1}{2 lim_{x \to 0} x + lim_{x \to 0} 1}}$

Step 4.6

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

$e^{\frac{1}{3} \cdot 2 \frac{1}{2 lim_{x \to 0} x + 1}}$

Step 5

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

$e^{\frac{1}{3} \cdot 2 \frac{1}{2 \cdot 0 + 1}}$

Step 6

Simplify the answer.

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Step 6.1

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

$e^{\frac{2}{3} \cdot \frac{1}{2 \cdot 0 + 1}}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Multiply $2$ by $0$ .

$e^{\frac{2}{3} \cdot \frac{1}{0 + 1}}$

Step 6.2.2

Add $0$ and $1$ .

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

Step 6.3

Cancel the common factor of $1$ .

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Step 6.3.1

Cancel the common factor.

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

Step 6.3.2

Rewrite the expression.

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

Step 6.4

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

$e^{\frac{2}{3}}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$e^{\frac{2}{3}}$

Decimal Form:

$1.94773404 \dots$