Calculus Examples

$lim_{x \to \infty} \frac{2^{x} - 5}{2^{x} + 7}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

$\frac{lim_{x \to \infty} 2^{x} - 5}{lim_{x \to \infty} 2^{x} + 7}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

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

$\frac{lim_{x \to \infty} 2^{x} - lim_{x \to \infty} 5}{lim_{x \to \infty} 2^{x} + 7}$

Step 1.1.2.2

Since the exponent $x$ approaches $\infty$ , the quantity $2^{x}$ approaches $\infty$ .

$\frac{\infty - lim_{x \to \infty} 5}{lim_{x \to \infty} 2^{x} + 7}$

Step 1.1.2.3

Evaluate the limit.

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Step 1.1.2.3.1

Evaluate the limit of $5$ which is constant as $x$ approaches $\infty$ .

$\frac{\infty - 1 \cdot 5}{lim_{x \to \infty} 2^{x} + 7}$

Step 1.1.2.3.2

Simplify the answer.

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Step 1.1.2.3.2.1

Multiply $- 1$ by $5$ .

$\frac{\infty - 5}{lim_{x \to \infty} 2^{x} + 7}$

Step 1.1.2.3.2.2

Infinity plus or minus a number is infinity.

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

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

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

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

Step 1.1.3.2

Since the exponent $x$ approaches $\infty$ , the quantity $2^{x}$ approaches $\infty$ .

$\frac{\infty}{\infty + lim_{x \to \infty} 7}$

Step 1.1.3.3

Evaluate the limit of $7$ which is constant as $x$ approaches $\infty$ .

$\frac{\infty}{\infty + 7}$

Step 1.1.3.4

Infinity plus or minus a number is infinity.

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

Step 1.1.3.5

Infinity divided by infinity is undefined.

Undefined

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

Step 1.1.4

Infinity divided by infinity is undefined.

Undefined

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

Step 1.2

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.

$lim_{x \to \infty} \frac{2^{x} - 5}{2^{x} + 7} = lim_{x \to \infty} \frac{\frac{d}{d x} [2^{x} - 5]}{\frac{d}{d x} [2^{x} + 7]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [2^{x} - 5]}{\frac{d}{d x} [2^{x} + 7]}$

Step 1.3.2

By the Sum Rule, the derivative of $2^{x} - 5$ with respect to $x$ is $\frac{d}{d x} [2^{x}] + \frac{d}{d x} [- 5]$ .

$lim_{x \to \infty} \frac{\frac{d}{d x} [2^{x}] + \frac{d}{d x} [- 5]}{\frac{d}{d x} [2^{x} + 7]}$

Step 1.3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $2$ .

$lim_{x \to \infty} \frac{2^{x} \ln (2) + \frac{d}{d x} [- 5]}{\frac{d}{d x} [2^{x} + 7]}$

Step 1.3.4

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

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

Step 1.3.5

Add $2^{x} \ln (2)$ and $0$ .

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

Step 1.3.6

By the Sum Rule, the derivative of $2^{x} + 7$ with respect to $x$ is $\frac{d}{d x} [2^{x}] + \frac{d}{d x} [7]$ .

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

Step 1.3.7

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $2$ .

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

Step 1.3.8

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$lim_{x \to \infty} \frac{2^{x} \ln (2)}{2^{x} \ln (2) + 0}$

Step 1.3.9

Add $2^{x} \ln (2)$ and $0$ .

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

Step 1.4

Reduce.

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Step 1.4.1

Cancel the common factor of $2^{x}$ .

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Step 1.4.1.1

Cancel the common factor.

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

Step 1.4.1.2

Rewrite the expression.

$lim_{x \to \infty} \frac{\ln (2)}{\ln (2)}$

Step 1.4.2

Cancel the common factor of $\ln (2)$ .

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Step 1.4.2.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\ln (2)}{\ln (2)}$

Step 1.4.2.2

Rewrite the expression.

$lim_{x \to \infty} 1$

Step 2

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

$1$