Calculus Examples

$lim_{x \to - 8} \frac{x}{\cos (3 x)}$

Step 1

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

$\frac{lim_{x \to - 8} x}{lim_{x \to - 8} \cos (3 x)}$

Step 2

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

$\frac{lim_{x \to - 8} x}{\cos (lim_{x \to - 8} 3 x)}$

Step 3

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{lim_{x \to - 8} x}{\cos (3 lim_{x \to - 8} x)}$

Step 4

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

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Step 4.1

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

$\frac{- 8}{\cos (3 lim_{x \to - 8} x)}$

Step 4.2

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

$\frac{- 8}{\cos (3 \cdot - 8)}$

Step 5

Simplify the answer.

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Step 5.1

Separate fractions.

$\frac{- 8}{1} \cdot \frac{1}{\cos (3 \cdot - 8)}$

Step 5.2

Convert from $\frac{1}{\cos (3 \cdot - 8)}$ to $\sec (3 \cdot - 8)$ .

$\frac{- 8}{1} \sec (3 \cdot - 8)$

Step 5.3

Divide $- 8$ by $1$ .

$- 8 \sec (3 \cdot - 8)$

Step 5.4

Multiply $3$ by $- 8$ .

$- 8 \sec (- 24)$

Step 5.5

Evaluate $\sec (- 24)$ .

$- 8 \cdot 1.09463627$

Step 5.6

Multiply $- 8$ by $1.09463627$ .

$- 8.75709022$