Calculus Examples

$lim_{x \to \infty} \frac{8 + 9 (6^{x})}{8 - 9 \cdot 6^{x}}$

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.

$\frac{lim_{x \to \infty} 8 + 9 (6^{x})}{lim_{x \to \infty} 8 - 9 \cdot 6^{x}}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

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

$\frac{lim_{x \to \infty} 8 + lim_{x \to \infty} 9 (6^{x})}{lim_{x \to \infty} 8 - 9 \cdot 6^{x}}$

Step 1.1.2.1.2

Evaluate the limit of $8$ which is constant as $x$ approaches $\infty$ .

$\frac{8 + lim_{x \to \infty} 9 (6^{x})}{lim_{x \to \infty} 8 - 9 \cdot 6^{x}}$

Step 1.1.2.2

Since the function $6^{x}$ approaches $\infty$ , the positive constant $9$ times the function also approaches $\infty$ .

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Step 1.1.2.2.1

Consider the limit with the constant multiple $9$ removed.

$\frac{8 + lim_{x \to \infty} 6^{x}}{lim_{x \to \infty} 8 - 9 \cdot 6^{x}}$

Step 1.1.2.2.2

Since the exponent $x$ approaches $\infty$ , the quantity $6^{x}$ approaches $\infty$ .

$\frac{8 + \infty}{lim_{x \to \infty} 8 - 9 \cdot 6^{x}}$

Step 1.1.2.3

Infinity plus or minus a number is infinity.

$\frac{\infty}{lim_{x \to \infty} 8 - 9 \cdot 6^{x}}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

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

$\frac{\infty}{lim_{x \to \infty} 8 - lim_{x \to \infty} 9 \cdot 6^{x}}$

Step 1.1.3.1.2

Evaluate the limit of $8$ which is constant as $x$ approaches $\infty$ .

$\frac{\infty}{8 - lim_{x \to \infty} 9 \cdot 6^{x}}$

Step 1.1.3.2

Since the function $6^{x}$ approaches $\infty$ , the positive constant $9$ times the function also approaches $\infty$ .

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Step 1.1.3.2.1

Consider the limit with the constant multiple $9$ removed.

$\frac{\infty}{8 - lim_{x \to \infty} 6^{x}}$

Step 1.1.3.2.2

Since the exponent $x$ approaches $\infty$ , the quantity $6^{x}$ approaches $\infty$ .

$\frac{\infty}{8 - \infty}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

A non-zero constant times infinity is infinity.

$\frac{\infty}{8 + - \infty}$

Step 1.1.3.3.2

Infinity plus or minus a number is infinity.

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

Step 1.1.3.3.3

Infinity divided by infinity is undefined.

Undefined

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

Step 1.1.3.4

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.

$lim_{x \to \infty} \frac{8 + 9 (6^{x})}{8 - 9 \cdot 6^{x}} = lim_{x \to \infty} \frac{\frac{d}{d x} [8 + 9 (6^{x})]}{\frac{d}{d x} [8 - 9 \cdot 6^{x}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [8 + 9 (6^{x})]}{\frac{d}{d x} [8 - 9 \cdot 6^{x}]}$

Step 1.3.2

By the Sum Rule, the derivative of $8 + 9 (6^{x})$ with respect to $x$ is $\frac{d}{d x} [8] + \frac{d}{d x} [9 (6^{x})]$ .

$lim_{x \to \infty} \frac{\frac{d}{d x} [8] + \frac{d}{d x} [9 (6^{x})]}{\frac{d}{d x} [8 - 9 \cdot 6^{x}]}$

Step 1.3.3

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$lim_{x \to \infty} \frac{0 + \frac{d}{d x} [9 (6^{x})]}{\frac{d}{d x} [8 - 9 \cdot 6^{x}]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [9 (6^{x})]$ .

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Step 1.3.4.1

Since $9$ is constant with respect to $x$ , the derivative of $9 \cdot 6^{x}$ with respect to $x$ is $9 \frac{d}{d x} [6^{x}]$ .

$lim_{x \to \infty} \frac{0 + 9 \frac{d}{d x} [6^{x}]}{\frac{d}{d x} [8 - 9 \cdot 6^{x}]}$

Step 1.3.4.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $6$ .

$lim_{x \to \infty} \frac{0 + 9 \cdot 6^{x} \ln (6)}{\frac{d}{d x} [8 - 9 \cdot 6^{x}]}$

Step 1.3.5

Add $0$ and $9 \cdot 6^{x} \ln (6)$ .

$lim_{x \to \infty} \frac{9 \cdot 6^{x} \ln (6)}{\frac{d}{d x} [8 - 9 \cdot 6^{x}]}$

Step 1.3.6

By the Sum Rule, the derivative of $8 - 9 \cdot 6^{x}$ with respect to $x$ is $\frac{d}{d x} [8] + \frac{d}{d x} [- 9 \cdot 6^{x}]$ .

$lim_{x \to \infty} \frac{9 \cdot 6^{x} \ln (6)}{\frac{d}{d x} [8] + \frac{d}{d x} [- 9 \cdot 6^{x}]}$

Step 1.3.7

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$lim_{x \to \infty} \frac{9 \cdot 6^{x} \ln (6)}{0 + \frac{d}{d x} [- 9 \cdot 6^{x}]}$

Step 1.3.8

Evaluate $\frac{d}{d x} [- 9 \cdot 6^{x}]$ .

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Step 1.3.8.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 \cdot 6^{x}$ with respect to $x$ is $- 9 \frac{d}{d x} [6^{x}]$ .

$lim_{x \to \infty} \frac{9 \cdot 6^{x} \ln (6)}{0 - 9 \frac{d}{d x} [6^{x}]}$

Step 1.3.8.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $6$ .

$lim_{x \to \infty} \frac{9 \cdot 6^{x} \ln (6)}{0 - 9 \cdot 6^{x} \ln (6)}$

Step 1.3.9

Subtract $9 \cdot 6^{x} \ln (6)$ from $0$ .

$lim_{x \to \infty} \frac{9 \cdot 6^{x} \ln (6)}{- 9 \cdot 6^{x} \ln (6)}$

Step 1.4

Reduce.

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Step 1.4.1

Cancel the common factor of $9$ and $- 9$ .

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Step 1.4.1.1

Factor $9$ out of $9 \cdot 6^{x} \ln (6)$ .

$lim_{x \to \infty} \frac{9 (6^{x} \ln (6))}{- 9 \cdot 6^{x} \ln (6)}$

Step 1.4.1.2

Cancel the common factors.

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Step 1.4.1.2.1

Factor $9$ out of $- 9 \cdot 6^{x} \ln (6)$ .

$lim_{x \to \infty} \frac{9 (6^{x} \ln (6))}{9 (- 6^{x} \ln (6))}$

Step 1.4.1.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{9 (6^{x} \ln (6))}{9 (- 6^{x} \ln (6))}$

Step 1.4.1.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{6^{x} \ln (6)}{- 6^{x} \ln (6)}$

Step 1.4.2

Cancel the common factor of $6^{x}$ .

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Step 1.4.2.1

Cancel the common factor.

$lim_{x \to \infty} \frac{6^{x} \ln (6)}{- 6^{x} \ln (6)}$

Step 1.4.2.2

Rewrite the expression.

$lim_{x \to \infty} \frac{\ln (6)}{- \ln (6)}$

Step 1.4.3

Cancel the common factor of $\ln (6)$ .

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Step 1.4.3.1

Cancel the common factor.

$lim_{x \to \infty} \frac{\ln (6)}{- \ln (6)}$

Step 1.4.3.2

Rewrite the expression.

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

Step 1.4.3.3

Move the negative one from the denominator of $\frac{1}{- 1}$ .

$lim_{x \to \infty} - 1 \cdot 1$

Step 2

Evaluate the limit.

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Step 2.1

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

$- 1 \cdot 1$

Step 2.2

Multiply $- 1$ by $1$ .

$- 1$