Calculus Examples

$lim_{x \to \infty} \frac{1 - e^{x}}{1 + 5 e^{x}}$

Step 1

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

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Step 1.1

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

$\frac{lim_{x \to \infty} 1 - e^{x}}{lim_{x \to \infty} 1 + 5 e^{x}}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Evaluate the limit.

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Step 1.2.1.1

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

$\frac{lim_{x \to \infty} 1 - lim_{x \to \infty} e^{x}}{lim_{x \to \infty} 1 + 5 e^{x}}$

Step 1.2.1.2

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

$\frac{1 - lim_{x \to \infty} e^{x}}{lim_{x \to \infty} 1 + 5 e^{x}}$

Step 1.2.2

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

$\frac{1 - \infty}{lim_{x \to \infty} 1 + 5 e^{x}}$

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

A non-zero constant times infinity is infinity.

$\frac{1 + - \infty}{lim_{x \to \infty} 1 + 5 e^{x}}$

Step 1.2.3.2

Infinity plus or minus a number is infinity.

$\frac{- \infty}{lim_{x \to \infty} 1 + 5 e^{x}}$

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Evaluate the limit.

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Step 1.3.1.1

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

$\frac{- \infty}{lim_{x \to \infty} 1 + lim_{x \to \infty} 5 e^{x}}$

Step 1.3.1.2

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

$\frac{- \infty}{1 + lim_{x \to \infty} 5 e^{x}}$

Step 1.3.2

Since the function $e^{x}$ approaches $\infty$ , the positive constant $5$ times the function also approaches $\infty$ .

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Step 1.3.2.1

Consider the limit with the constant multiple $5$ removed.

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

Step 1.3.2.2

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

$\frac{- \infty}{1 + \infty}$

Step 1.3.3

Infinity plus or minus a number is infinity.

$\frac{- \infty}{\infty}$

Step 1.3.4

Infinity divided by infinity is undefined.

Undefined

$\frac{- \infty}{\infty}$

Step 1.4

Infinity divided by infinity is undefined.

Undefined

$\frac{- \infty}{\infty}$

Step 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.

$lim_{x \to \infty} \frac{1 - e^{x}}{1 + 5 e^{x}} = lim_{x \to \infty} \frac{\frac{d}{d x} [1 - e^{x}]}{\frac{d}{d x} [1 + 5 e^{x}]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [1 - e^{x}]}{\frac{d}{d x} [1 + 5 e^{x}]}$

Step 3.2

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

$lim_{x \to \infty} \frac{\frac{d}{d x} [1] + \frac{d}{d x} [- e^{x}]}{\frac{d}{d x} [1 + 5 e^{x}]}$

Step 3.3

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

$lim_{x \to \infty} \frac{0 + \frac{d}{d x} [- e^{x}]}{\frac{d}{d x} [1 + 5 e^{x}]}$

Step 3.4

Evaluate $\frac{d}{d x} [- e^{x}]$ .

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Step 3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- e^{x}$ with respect to $x$ is $- \frac{d}{d x} [e^{x}]$ .

$lim_{x \to \infty} \frac{0 - \frac{d}{d x} [e^{x}]}{\frac{d}{d x} [1 + 5 e^{x}]}$

Step 3.4.2

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

$lim_{x \to \infty} \frac{0 - e^{x}}{\frac{d}{d x} [1 + 5 e^{x}]}$

Step 3.5

Subtract $e^{x}$ from $0$ .

$lim_{x \to \infty} \frac{- e^{x}}{\frac{d}{d x} [1 + 5 e^{x}]}$

Step 3.6

By the Sum Rule, the derivative of $1 + 5 e^{x}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [5 e^{x}]$ .

$lim_{x \to \infty} \frac{- e^{x}}{\frac{d}{d x} [1] + \frac{d}{d x} [5 e^{x}]}$

Step 3.7

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

$lim_{x \to \infty} \frac{- e^{x}}{0 + \frac{d}{d x} [5 e^{x}]}$

Step 3.8

Evaluate $\frac{d}{d x} [5 e^{x}]$ .

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Step 3.8.1

Since $5$ is constant with respect to $x$ , the derivative of $5 e^{x}$ with respect to $x$ is $5 \frac{d}{d x} [e^{x}]$ .

$lim_{x \to \infty} \frac{- e^{x}}{0 + 5 \frac{d}{d x} [e^{x}]}$

Step 3.8.2

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

$lim_{x \to \infty} \frac{- e^{x}}{0 + 5 e^{x}}$

Step 3.9

Add $0$ and $5 e^{x}$ .

$lim_{x \to \infty} \frac{- e^{x}}{5 e^{x}}$

Step 4

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

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Step 4.1

Cancel the common factor.

$lim_{x \to \infty} \frac{- e^{x}}{5 e^{x}}$

Step 4.2

Rewrite the expression.

$lim_{x \to \infty} \frac{- 1}{5}$

Step 5

Evaluate the limit of $\frac{- 1}{5}$ which is constant as $x$ approaches $\infty$ .

$\frac{- 1}{5}$

Step 6

Move the negative in front of the fraction.

$- \frac{1}{5}$