Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Evaluate the limit.

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Step 2.1

Move the limit into the exponent.

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

Step 2.2

Combine $\frac{1}{x}$ and $\ln (e^{x} + x)$ .

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

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.

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

Step 3.1.2

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

$e^{\frac{\infty}{lim_{x \to \infty} x}}$

Step 3.1.3

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

$e^{\frac{\infty}{\infty}}$

Step 3.1.4

Infinity divided by infinity is undefined.

Undefined

$e^{\frac{\infty}{\infty}}$

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.

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

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

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 $g (x) = e^{x} + x$ .

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Step 3.3.2.1

To apply the Chain Rule, set $u$ as $e^{x} + x$ .

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

Step 3.3.2.2

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

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

Step 3.3.2.3

Replace all occurrences of $u$ with $e^{x} + x$ .

$e^{lim_{x \to \infty} \frac{\frac{1}{e^{x} + x} \frac{d}{d x} [e^{x} + x]}{\frac{d}{d x} [x]}}$

Step 3.3.3

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

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

Step 3.3.4

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

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

Step 3.3.5

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

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

Step 3.3.6

Simplify.

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Step 3.3.6.1

Reorder the factors of $\frac{1}{e^{x} + x} (e^{x} + 1)$ .

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

Step 3.3.6.2

Multiply $e^{x} + 1$ by $\frac{1}{e^{x} + x}$ .

$e^{lim_{x \to \infty} \frac{\frac{e^{x} + 1}{e^{x} + x}}{\frac{d}{d x} [x]}}$

Step 3.3.7

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

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

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.5

Multiply $\frac{e^{x} + 1}{e^{x} + x}$ by $1$ .

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

Step 4

Apply L'Hospital's rule.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

Evaluate the limit of the numerator.

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Step 4.1.2.1

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

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

Step 4.1.2.2

Since the exponent $x$ approaches $\infty$ , the quantity $e^{x}$ approaches $\infty$ .

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

Step 4.1.2.3

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

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

Step 4.1.2.4

Infinity plus or minus a number is infinity.

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

Step 4.1.3

Evaluate the limit of the denominator.

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Step 4.1.3.1

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

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

Step 4.1.3.2

Since the exponent $x$ approaches $\infty$ , the quantity $e^{x}$ approaches $\infty$ .

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

Step 4.1.3.3

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

$e^{\frac{\infty}{\infty + \infty}}$

Step 4.1.3.4

Infinity plus infinity is infinity.

$e^{\frac{\infty}{\infty}}$

Step 4.1.3.5

Infinity divided by infinity is undefined.

Undefined

$e^{\frac{\infty}{\infty}}$

Step 4.1.4

Infinity divided by infinity is undefined.

Undefined

$e^{\frac{\infty}{\infty}}$

Step 4.2

$lim_{x \to \infty} \frac{e^{x} + 1}{e^{x} + x} = lim_{x \to \infty} \frac{\frac{d}{d x} [e^{x} + 1]}{\frac{d}{d x} [e^{x} + x]}$

Step 4.3

Find the derivative of the numerator and denominator.

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Step 4.3.1

Differentiate the numerator and denominator.

$e^{lim_{x \to \infty} \frac{\frac{d}{d x} [e^{x} + 1]}{\frac{d}{d x} [e^{x} + x]}}$

Step 4.3.2

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

$e^{lim_{x \to \infty} \frac{\frac{d}{d x} [e^{x}] + \frac{d}{d x} [1]}{\frac{d}{d x} [e^{x} + x]}}$

Step 4.3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$e^{lim_{x \to \infty} \frac{e^{x} + \frac{d}{d x} [1]}{\frac{d}{d x} [e^{x} + x]}}$

Step 4.3.4

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

$e^{lim_{x \to \infty} \frac{e^{x} + 0}{\frac{d}{d x} [e^{x} + x]}}$

Step 4.3.5

Add $e^{x}$ and $0$ .

$e^{lim_{x \to \infty} \frac{e^{x}}{\frac{d}{d x} [e^{x} + x]}}$

Step 4.3.6

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

$e^{lim_{x \to \infty} \frac{e^{x}}{\frac{d}{d x} [e^{x}] + \frac{d}{d x} [x]}}$

Step 4.3.7

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$e^{lim_{x \to \infty} \frac{e^{x}}{e^{x} + \frac{d}{d x} [x]}}$

Step 4.3.8

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

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

Step 5

Apply L'Hospital's rule.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Since the exponent $x$ approaches $\infty$ , the quantity $e^{x}$ approaches $\infty$ .

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

Step 5.1.3

Evaluate the limit of the denominator.

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Step 5.1.3.1

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

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

Step 5.1.3.2

Since the exponent $x$ approaches $\infty$ , the quantity $e^{x}$ approaches $\infty$ .

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

Step 5.1.3.3

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

$e^{\frac{\infty}{\infty + 1}}$

Step 5.1.3.4

Infinity plus or minus a number is infinity.

$e^{\frac{\infty}{\infty}}$

Step 5.1.3.5

Infinity divided by infinity is undefined.

Undefined

$e^{\frac{\infty}{\infty}}$

Step 5.1.4

Infinity divided by infinity is undefined.

Undefined

$e^{\frac{\infty}{\infty}}$

Step 5.2

$lim_{x \to \infty} \frac{e^{x}}{e^{x} + 1} = lim_{x \to \infty} \frac{\frac{d}{d x} [e^{x}]}{\frac{d}{d x} [e^{x} + 1]}$

Step 5.3

Find the derivative of the numerator and denominator.

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Step 5.3.1

Differentiate the numerator and denominator.

$e^{lim_{x \to \infty} \frac{\frac{d}{d x} [e^{x}]}{\frac{d}{d x} [e^{x} + 1]}}$

Step 5.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$e^{lim_{x \to \infty} \frac{e^{x}}{\frac{d}{d x} [e^{x} + 1]}}$

Step 5.3.3

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

$e^{lim_{x \to \infty} \frac{e^{x}}{\frac{d}{d x} [e^{x}] + \frac{d}{d x} [1]}}$

Step 5.3.4

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$e^{lim_{x \to \infty} \frac{e^{x}}{e^{x} + \frac{d}{d x} [1]}}$

Step 5.3.5

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

$e^{lim_{x \to \infty} \frac{e^{x}}{e^{x} + 0}}$

Step 5.3.6

Add $e^{x}$ and $0$ .

$e^{lim_{x \to \infty} \frac{e^{x}}{e^{x}}}$

Step 5.4

Cancel the common factor of $e^{x}$ .

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Step 5.4.1

Cancel the common factor.

$e^{lim_{x \to \infty} \frac{e^{x}}{e^{x}}}$

Step 5.4.2

Rewrite the expression.

$e^{lim_{x \to \infty} 1}$

Step 6

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

$e^{1}$

Step 7

Simplify.

$e$

Step 8

The result can be shown in multiple forms.

Exact Form:

$e$

Decimal Form:

$2.71828182 \dots$