Calculus Examples

$lim_{x \to \infty} \frac{2 - 15 e^{- 5 x}}{\frac{1}{x} + 4 \cos (\frac{5}{x})}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{lim_{x \to \infty} 2 - 15 e^{- 5 x}}{lim_{x \to \infty} \frac{1}{x} + 4 \cos (\frac{5}{x})}$

Step 1.2

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

$\frac{lim_{x \to \infty} 2 - lim_{x \to \infty} 15 e^{- 5 x}}{lim_{x \to \infty} \frac{1}{x} + 4 \cos (\frac{5}{x})}$

Step 1.3

Evaluate the limit of $2$ which is constant as $x$ approaches $\infty$ .

$\frac{2 - lim_{x \to \infty} 15 e^{- 5 x}}{lim_{x \to \infty} \frac{1}{x} + 4 \cos (\frac{5}{x})}$

Step 1.4

Move the term $15$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 - 15 lim_{x \to \infty} e^{- 5 x}}{lim_{x \to \infty} \frac{1}{x} + 4 \cos (\frac{5}{x})}$

Step 2

Since the exponent $- 5 x$ approaches $- \infty$ , the quantity $e^{- 5 x}$ approaches $0$ .

$\frac{2 - 15 \cdot 0}{lim_{x \to \infty} \frac{1}{x} + 4 \cos (\frac{5}{x})}$

Step 3

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

$\frac{2 - 15 \cdot 0}{lim_{x \to \infty} \frac{1}{x} + lim_{x \to \infty} 4 \cos (\frac{5}{x})}$

Step 4

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$\frac{2 - 15 \cdot 0}{0 + lim_{x \to \infty} 4 \cos (\frac{5}{x})}$

Step 5

Evaluate the limit.

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Step 5.1

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 - 15 \cdot 0}{0 + 4 lim_{x \to \infty} \cos (\frac{5}{x})}$

Step 5.2

Move the limit inside the trig function because cosine is continuous.

$\frac{2 - 15 \cdot 0}{0 + 4 \cos (lim_{x \to \infty} \frac{5}{x})}$

Step 5.3

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 - 15 \cdot 0}{0 + 4 \cos (5 lim_{x \to \infty} \frac{1}{x})}$

Step 6

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{x}$ approaches $0$ .

$\frac{2 - 15 \cdot 0}{0 + 4 \cos (5 \cdot 0)}$

Step 7

Simplify the answer.

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Step 7.1

Cancel the common factor of $2 - 15 \cdot 0$ and $0 + 4 \cos (5 \cdot 0)$ .

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Step 7.1.1

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{- 1 (- 2) - 15 \cdot 0}{0 + 4 \cos (5 \cdot 0)}$

Step 7.1.2

Factor $- 1$ out of $- 15 \cdot 0$ .

$\frac{- 1 (- 2) - (15 \cdot 0)}{0 + 4 \cos (5 \cdot 0)}$

Step 7.1.3

Factor $- 1$ out of $- 1 (- 2) - (15 \cdot 0)$ .

$\frac{- 1 (- 2 + 15 \cdot 0)}{0 + 4 \cos (5 \cdot 0)}$

Step 7.1.4

Reorder terms.

$\frac{- 1 (- 2 + 0 \cdot 15)}{0 + 4 \cos (5 \cdot 0)}$

Step 7.1.5

Factor $2$ out of $- 1 (- 2 + 0 \cdot 15)$ .

$\frac{2 (- 1 (- 1 + 0 \cdot 15))}{0 + 4 \cos (5 \cdot 0)}$

Step 7.1.6

Cancel the common factors.

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Step 7.1.6.1

Factor $2$ out of $0$ .

$\frac{2 (- 1 (- 1 + 0 \cdot 15))}{2 (0) + 4 \cos (5 \cdot 0)}$

Step 7.1.6.2

Factor $2$ out of $4 \cos (5 \cdot 0)$ .

$\frac{2 (- 1 (- 1 + 0 \cdot 15))}{2 (0) + 2 (2 \cos (5 \cdot 0))}$

Step 7.1.6.3

Factor $2$ out of $2 (0) + 2 (2 \cos (5 \cdot 0))$ .

$\frac{2 (- 1 (- 1 + 0 \cdot 15))}{2 (0 + 2 \cos (5 \cdot 0))}$

Step 7.1.6.4

Cancel the common factor.

$\frac{2 (- 1 (- 1 + 0 \cdot 15))}{2 (0 + 2 \cos (5 \cdot 0))}$

Step 7.1.6.5

Rewrite the expression.

$\frac{- 1 (- 1 + 0 \cdot 15)}{0 + 2 \cos (5 \cdot 0)}$

Step 7.2

Simplify the numerator.

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Step 7.2.1

Multiply $0$ by $15$ .

$\frac{- 1 (- 1 + 0)}{0 + 2 \cos (5 \cdot 0)}$

Step 7.2.2

Add $- 1$ and $0$ .

$\frac{- 1 \cdot - 1}{0 + 2 \cos (5 \cdot 0)}$

Step 7.3

Simplify the denominator.

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Step 7.3.1

Multiply $5$ by $0$ .

$\frac{- 1 \cdot - 1}{0 + 2 \cos (0)}$

Step 7.3.2

The exact value of $\cos (0)$ is $1$ .

$\frac{- 1 \cdot - 1}{0 + 2 \cdot 1}$

Step 7.3.3

Multiply $2$ by $1$ .

$\frac{- 1 \cdot - 1}{0 + 2}$

Step 7.3.4

Add $0$ and $2$ .

$\frac{- 1 \cdot - 1}{2}$

Step 7.4

Multiply $- 1$ by $- 1$ .

$\frac{1}{2}$