Calculus Examples

$lim_{x \to \infty} \frac{1}{x} + 4 \cos (\frac{5}{x})$

Step 1

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

$lim_{x \to \infty} \frac{1}{x} + lim_{x \to \infty} 4 \cos (\frac{5}{x})$

Step 2

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

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

Step 3

Evaluate the limit.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

$0 + 4 \cos (5 \cdot 0)$

Step 5

Simplify the answer.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $5$ by $0$ .

$0 + 4 \cos (0)$

Step 5.1.2

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

$0 + 4 \cdot 1$

Step 5.1.3

Multiply $4$ by $1$ .

$0 + 4$

Step 5.2

Add $0$ and $4$ .

$4$