Calculus Examples

$lim_{x \to 8} \cos (\frac{5 x}{x^{2} + 7} + \frac{π}{3})$

Step 1

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

$\cos (lim_{x \to 8} \frac{5 x}{x^{2} + 7} + \frac{π}{3})$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\cos (lim_{x \to 8} \frac{5 x}{x^{2} + 7} + lim_{x \to 8} \frac{π}{3})$

Step 3

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

$\cos (5 lim_{x \to 8} \frac{x}{x^{2} + 7} + lim_{x \to 8} \frac{π}{3})$

Step 4

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$\cos (5 \frac{lim_{x \to 8} x}{lim_{x \to 8} x^{2} + 7} + lim_{x \to 8} \frac{π}{3})$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\cos (5 \frac{lim_{x \to 8} x}{lim_{x \to 8} x^{2} + lim_{x \to 8} 7} + lim_{x \to 8} \frac{π}{3})$

Step 6

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\cos (5 \frac{lim_{x \to 8} x}{{(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 7} + lim_{x \to 8} \frac{π}{3})$

Step 7

Evaluate the limit of $7$ which is constant as $x$ approaches $8$ .

$\cos (5 \frac{lim_{x \to 8} x}{{(lim_{x \to 8} x)}^{2} + 7} + lim_{x \to 8} \frac{π}{3})$

Step 8

Evaluate the limit of $\frac{π}{3}$ which is constant as $x$ approaches $8$ .

$\cos (5 \frac{lim_{x \to 8} x}{{(lim_{x \to 8} x)}^{2} + 7} + \frac{π}{3})$

Step 9

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

Tap for more steps...

Step 9.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\cos (5 \frac{8}{{(lim_{x \to 8} x)}^{2} + 7} + \frac{π}{3})$

Step 9.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\cos (5 \frac{8}{8^{2} + 7} + \frac{π}{3})$

Step 10

Simplify the answer.

Tap for more steps...

Step 10.1

Simplify each term.

Tap for more steps...

Step 10.1.1

Simplify the denominator.

Tap for more steps...

Step 10.1.1.1

Raise $8$ to the power of $2$ .

$\cos (5 \frac{8}{64 + 7} + \frac{π}{3})$

Step 10.1.1.2

Add $64$ and $7$ .

$\cos (5 (\frac{8}{71}) + \frac{π}{3})$

Step 10.1.2

Multiply $5 (\frac{8}{71})$ .

Tap for more steps...

Step 10.1.2.1

Combine $5$ and $\frac{8}{71}$ .

$\cos (\frac{5 \cdot 8}{71} + \frac{π}{3})$

Step 10.1.2.2

Multiply $5$ by $8$ .

$\cos (\frac{40}{71} + \frac{π}{3})$

Step 10.2

Evaluate $\cos (\frac{40}{71} + \frac{π}{3})$ .

$- 0.03977101$