Calculus Examples

$lim_{x \to \infty} {(1 + 2 x)}^{\frac{7}{2 \ln (x)}}$

Step 1

Use the properties of logarithms to simplify the limit.

Tap for more steps...

Step 1.1

Rewrite ${(1 + 2 x)}^{\frac{7}{2 \ln (x)}}$ as $e^{\ln ({(1 + 2 x)}^{\frac{7}{2 \ln (x)}})}$ .

$lim_{x \to \infty} e^{\ln ({(1 + 2 x)}^{\frac{7}{2 \ln (x)}})}$

Step 1.2

Expand $\ln ({(1 + 2 x)}^{\frac{7}{2 \ln (x)}})$ by moving $\frac{7}{2 \ln (x)}$ outside the logarithm.

$lim_{x \to \infty} e^{\frac{7}{2 \ln (x)} \ln (1 + 2 x)}$

Step 2

Evaluate the limit.

Tap for more steps...

Step 2.1

Move the limit into the exponent.

$e^{lim_{x \to \infty} \frac{7}{2 \ln (x)} \ln (1 + 2 x)}$

Step 2.2

Combine $\frac{7}{2 \ln (x)}$ and $\ln (1 + 2 x)$ .

$e^{lim_{x \to \infty} \frac{7 \ln (1 + 2 x)}{2 \ln (x)}}$

Step 2.3

Move the term $\frac{7}{2}$ outside of the limit because it is constant with respect to $x$ .

$e^{\frac{7}{2} lim_{x \to \infty} \frac{\ln (1 + 2 x)}{\ln (x)}}$

Step 3

Apply L'Hospital's rule.

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

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

$e^{\frac{7}{2} \cdot \frac{lim_{x \to \infty} \ln (1 + 2 x)}{lim_{x \to \infty} \ln (x)}}$

Step 3.1.2

As log approaches infinity, the value goes to $\infty$ .

$e^{\frac{7}{2} \cdot \frac{\infty}{lim_{x \to \infty} \ln (x)}}$

Step 3.1.3

As log approaches infinity, the value goes to $\infty$ .

$e^{\frac{7}{2} \cdot \frac{\infty}{\infty}}$

Step 3.1.4

Infinity divided by infinity is undefined.

Undefined

$e^{\frac{7}{2} \cdot \frac{\infty}{\infty}}$

Step 3.2

Since $\frac{\infty}{\infty}$ 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.

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

Step 3.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 3.3.1

Differentiate the numerator and denominator.

$e^{\frac{7}{2} lim_{x \to \infty} \frac{\frac{d}{d x} [\ln (1 + 2 x)]}{\frac{d}{d x} [\ln (x)]}}$

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

Tap for more steps...

Step 3.3.2.1

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

$e^{\frac{7}{2} lim_{x \to \infty} \frac{\frac{d}{d u} [\ln (u)] \frac{d}{d x} [1 + 2 x]}{\frac{d}{d x} [\ln (x)]}}$

Step 3.3.2.2

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

$e^{\frac{7}{2} lim_{x \to \infty} \frac{\frac{1}{u} \frac{d}{d x} [1 + 2 x]}{\frac{d}{d x} [\ln (x)]}}$

Step 3.3.2.3

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

$e^{\frac{7}{2} lim_{x \to \infty} \frac{\frac{1}{1 + 2 x} \frac{d}{d x} [1 + 2 x]}{\frac{d}{d x} [\ln (x)]}}$

Step 3.3.3

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

$e^{\frac{7}{2} lim_{x \to \infty} \frac{\frac{1}{1 + 2 x} (\frac{d}{d x} [1] + \frac{d}{d x} [2 x])}{\frac{d}{d x} [\ln (x)]}}$

Step 3.3.4

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

$e^{\frac{7}{2} lim_{x \to \infty} \frac{\frac{1}{1 + 2 x} (0 + \frac{d}{d x} [2 x])}{\frac{d}{d x} [\ln (x)]}}$

Step 3.3.5

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

$e^{\frac{7}{2} lim_{x \to \infty} \frac{\frac{1}{1 + 2 x} \frac{d}{d x} [2 x]}{\frac{d}{d x} [\ln (x)]}}$

Step 3.3.6

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

$e^{\frac{7}{2} lim_{x \to \infty} \frac{\frac{1}{1 + 2 x} \cdot 2 \frac{d}{d x} [x]}{\frac{d}{d x} [\ln (x)]}}$

Step 3.3.7

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

$e^{\frac{7}{2} lim_{x \to \infty} \frac{\frac{2}{1 + 2 x} \frac{d}{d x} [x]}{\frac{d}{d x} [\ln (x)]}}$

Step 3.3.8

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

$e^{\frac{7}{2} lim_{x \to \infty} \frac{\frac{2}{1 + 2 x} \cdot 1}{\frac{d}{d x} [\ln (x)]}}$

Step 3.3.9

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

$e^{\frac{7}{2} lim_{x \to \infty} \frac{\frac{2}{1 + 2 x}}{\frac{d}{d x} [\ln (x)]}}$

Step 3.3.10

Reorder terms.

$e^{\frac{7}{2} lim_{x \to \infty} \frac{\frac{2}{2 x + 1}}{\frac{d}{d x} [\ln (x)]}}$

Step 3.3.11

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

$e^{\frac{7}{2} lim_{x \to \infty} \frac{\frac{2}{2 x + 1}}{\frac{1}{x}}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$e^{\frac{7}{2} lim_{x \to \infty} \frac{2}{2 x + 1} x}$

Step 3.5

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

$e^{\frac{7}{2} lim_{x \to \infty} \frac{2 x}{2 x + 1}}$

Step 4

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

$e^{\frac{7}{2} \cdot 2 lim_{x \to \infty} \frac{x}{2 x + 1}}$

Step 5

Divide the numerator and denominator by the highest power of $x$ in the denominator, which is $x$ .

$e^{\frac{7}{2} \cdot 2 lim_{x \to \infty} \frac{\frac{x}{x}}{\frac{2 x}{x} + \frac{1}{x}}}$

Step 6

Evaluate the limit.

Tap for more steps...

Step 6.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 6.1.1

Cancel the common factor.

$e^{\frac{7}{2} \cdot 2 lim_{x \to \infty} \frac{\frac{x}{x}}{\frac{2 x}{x} + \frac{1}{x}}}$

Step 6.1.2

Rewrite the expression.

$e^{\frac{7}{2} \cdot 2 lim_{x \to \infty} \frac{1}{\frac{2 x}{x} + \frac{1}{x}}}$

Step 6.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 6.2.1

Cancel the common factor.

$e^{\frac{7}{2} \cdot 2 lim_{x \to \infty} \frac{1}{\frac{2 x}{x} + \frac{1}{x}}}$

Step 6.2.2

Divide $2$ by $1$ .

$e^{\frac{7}{2} \cdot 2 lim_{x \to \infty} \frac{1}{2 + \frac{1}{x}}}$

Step 6.3

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

$e^{\frac{7}{2} \cdot 2 \frac{lim_{x \to \infty} 1}{lim_{x \to \infty} 2 + \frac{1}{x}}}$

Step 6.4

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

$e^{\frac{7}{2} \cdot 2 \frac{1}{lim_{x \to \infty} 2 + \frac{1}{x}}}$

Step 6.5

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

$e^{\frac{7}{2} \cdot 2 \frac{1}{lim_{x \to \infty} 2 + lim_{x \to \infty} \frac{1}{x}}}$

Step 6.6

Evaluate the limit of $2$ which is constant as $x$ approaches $\infty$ .

$e^{\frac{7}{2} \cdot 2 \frac{1}{2 + lim_{x \to \infty} \frac{1}{x}}}$

Step 7

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$e^{\frac{7}{2} \cdot 2 \frac{1}{2 + 0}}$

Step 8

Simplify the answer.

Tap for more steps...

Step 8.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.1.1

Cancel the common factor.

$e^{\frac{7}{2} \cdot 2 \frac{1}{2 + 0}}$

Step 8.1.2

Rewrite the expression.

$e^{7 \frac{1}{2 + 0}}$

Step 8.2

Add $2$ and $0$ .

$e^{7 (\frac{1}{2})}$

Step 8.3

Combine $7$ and $\frac{1}{2}$ .

$e^{\frac{7}{2}}$