Calculus Examples

$lim_{x \to \infty} \frac{3 e^{x}}{9 + 2 e^{5 x}}$

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.

$\frac{lim_{x \to \infty} 3 e^{x}}{lim_{x \to \infty} 9 + 2 e^{5 x}}$

Step 1.2

Since the function $e^{x}$ approaches $\infty$ , the positive constant $3$ times the function also approaches $\infty$ .

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Step 1.2.1

Consider the limit with the constant multiple $3$ removed.

$\frac{lim_{x \to \infty} e^{x}}{lim_{x \to \infty} 9 + 2 e^{5 x}}$

Step 1.2.2

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

$\frac{\infty}{lim_{x \to \infty} 9 + 2 e^{5 x}}$

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 $\infty$ .

$\frac{\infty}{lim_{x \to \infty} 9 + lim_{x \to \infty} 2 e^{5 x}}$

Step 1.3.1.2

Evaluate the limit of $9$ which is constant as $x$ approaches $\infty$ .

$\frac{\infty}{9 + lim_{x \to \infty} 2 e^{5 x}}$

Step 1.3.2

Since the function $e^{5 x}$ approaches $\infty$ , the positive constant $2$ times the function also approaches $\infty$ .

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Step 1.3.2.1

Consider the limit with the constant multiple $2$ removed.

$\frac{\infty}{9 + lim_{x \to \infty} e^{5 x}}$

Step 1.3.2.2

Since the exponent $5 x$ approaches $\infty$ , the quantity $e^{5 x}$ approaches $\infty$ .

$\frac{\infty}{9 + \infty}$

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.

$lim_{x \to \infty} \frac{3 e^{x}}{9 + 2 e^{5 x}} = lim_{x \to \infty} \frac{\frac{d}{d x} [3 e^{x}]}{\frac{d}{d x} [9 + 2 e^{5 x}]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to \infty} \frac{\frac{d}{d x} [3 e^{x}]}{\frac{d}{d x} [9 + 2 e^{5 x}]}$

Step 3.2

Since $3$ is constant with respect to $x$ , the derivative of $3 e^{x}$ with respect to $x$ is $3 \frac{d}{d x} [e^{x}]$ .

$lim_{x \to \infty} \frac{3 \frac{d}{d x} [e^{x}]}{\frac{d}{d x} [9 + 2 e^{5 x}]}$

Step 3.3

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

$lim_{x \to \infty} \frac{3 e^{x}}{\frac{d}{d x} [9 + 2 e^{5 x}]}$

Step 3.4

By the Sum Rule, the derivative of $9 + 2 e^{5 x}$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [2 e^{5 x}]$ .

$lim_{x \to \infty} \frac{3 e^{x}}{\frac{d}{d x} [9] + \frac{d}{d x} [2 e^{5 x}]}$

Step 3.5

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$lim_{x \to \infty} \frac{3 e^{x}}{0 + \frac{d}{d x} [2 e^{5 x}]}$

Step 3.6

Evaluate $\frac{d}{d x} [2 e^{5 x}]$ .

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Step 3.6.1

Since $2$ is constant with respect to $x$ , the derivative of $2 e^{5 x}$ with respect to $x$ is $2 \frac{d}{d x} [e^{5 x}]$ .

$lim_{x \to \infty} \frac{3 e^{x}}{0 + 2 \frac{d}{d x} [e^{5 x}]}$

Step 3.6.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 5 x$ .

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Step 3.6.2.1

To apply the Chain Rule, set $u$ as $5 x$ .

$lim_{x \to \infty} \frac{3 e^{x}}{0 + 2 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [5 x])}$

Step 3.6.2.2

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

$lim_{x \to \infty} \frac{3 e^{x}}{0 + 2 (e^{u} \frac{d}{d x} [5 x])}$

Step 3.6.2.3

Replace all occurrences of $u$ with $5 x$ .

$lim_{x \to \infty} \frac{3 e^{x}}{0 + 2 (e^{5 x} \frac{d}{d x} [5 x])}$

Step 3.6.3

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

$lim_{x \to \infty} \frac{3 e^{x}}{0 + 2 (e^{5 x} (5 \frac{d}{d x} [x]))}$

Step 3.6.4

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

$lim_{x \to \infty} \frac{3 e^{x}}{0 + 2 (e^{5 x} (5 \cdot 1))}$

Step 3.6.5

Multiply $5$ by $1$ .

$lim_{x \to \infty} \frac{3 e^{x}}{0 + 2 (e^{5 x} \cdot 5)}$

Step 3.6.6

Move $5$ to the left of $e^{5 x}$ .

$lim_{x \to \infty} \frac{3 e^{x}}{0 + 2 (5 e^{5 x})}$

Step 3.6.7

Multiply $5$ by $2$ .

$lim_{x \to \infty} \frac{3 e^{x}}{0 + 10 e^{5 x}}$

Step 3.7

Add $0$ and $10 e^{5 x}$ .

$lim_{x \to \infty} \frac{3 e^{x}}{10 e^{5 x}}$

Step 4

Evaluate the limit.

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Step 4.1

Cancel the common factor of $e^{x}$ and $e^{5 x}$ .

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Step 4.1.1

Factor $e^{5 x}$ out of $3 e^{x}$ .

$lim_{x \to \infty} \frac{e^{5 x} (3 e^{- 4 x})}{10 e^{5 x}}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $e^{5 x}$ out of $10 e^{5 x}$ .

$lim_{x \to \infty} \frac{e^{5 x} (3 e^{- 4 x})}{e^{5 x} \cdot 10}$

Step 4.1.2.2

Cancel the common factor.

$lim_{x \to \infty} \frac{e^{5 x} (3 e^{- 4 x})}{e^{5 x} \cdot 10}$

Step 4.1.2.3

Rewrite the expression.

$lim_{x \to \infty} \frac{3 e^{- 4 x}}{10}$

Step 4.2

Move the term $\frac{3}{10}$ outside of the limit because it is constant with respect to $x$ .

$\frac{3}{10} lim_{x \to \infty} e^{- 4 x}$

Step 5

Since the exponent $- 4 x$ approaches $- \infty$ , the quantity $e^{- 4 x}$ approaches $0$ .

$\frac{3}{10} \cdot 0$

Step 6

Multiply $\frac{3}{10}$ by $0$ .

$0$