Calculus Examples

$lim_{x \to \infty} \cos (\frac{1}{x}) - 1^{x}$

Step 1

Evaluate the limit.

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Step 1.1

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

$lim_{x \to \infty} \cos (\frac{1}{x}) - lim_{x \to \infty} 1^{x}$

Step 1.2

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

$\cos (lim_{x \to \infty} \frac{1}{x}) - lim_{x \to \infty} 1^{x}$

Step 2

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

$\cos (0) - lim_{x \to \infty} 1^{x}$

Step 3

Raising $1$ to any power yields $1$ .

$\cos (0) - 1 \cdot 1$

Step 4

Simplify the answer.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

$1 - 1 \cdot 1$

Step 4.1.2

Multiply $- 1$ by $1$ .

$1 - 1$

Step 4.2

Subtract $1$ from $1$ .

$0$