Calculus Examples

$lim_{x \to 3} \cos (\frac{1}{x} - \frac{1}{3})$

Step 1

Evaluate the limit.

Tap for more steps...

Step 1.1

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

$\cos (lim_{x \to 3} \frac{1}{x} - \frac{1}{3})$

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $3$ .

$\cos (lim_{x \to 3} \frac{1}{x} - lim_{x \to 3} \frac{1}{3})$

Step 1.3

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

$\cos (\frac{lim_{x \to 3} 1}{lim_{x \to 3} x} - lim_{x \to 3} \frac{1}{3})$

Step 1.4

Evaluate the limit of $1$ which is constant as $x$ approaches $3$ .

$\cos (\frac{1}{lim_{x \to 3} x} - lim_{x \to 3} \frac{1}{3})$

Step 1.5

Evaluate the limit of $\frac{1}{3}$ which is constant as $x$ approaches $3$ .

$\cos (\frac{1}{lim_{x \to 3} x} - \frac{1}{3})$

Step 2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

$\cos (\frac{1}{3} - \frac{1}{3})$

Step 3

Simplify the answer.

Tap for more steps...

Step 3.1

Combine the numerators over the common denominator.

$\cos (\frac{1 - 1}{3})$

Step 3.2

Subtract $1$ from $1$ .

$\cos (\frac{0}{3})$

Step 3.3

Divide $0$ by $3$ .

$\cos (0)$

Step 3.4

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

$1$