Calculus Examples

$lim_{x \to - 8} \frac{9 - 8 x + \sin (2 x)}{9 x + \cos (2 x)}$

Step 1

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

$\frac{lim_{x \to - 8} 9 - 8 x + \sin (2 x)}{lim_{x \to - 8} 9 x + \cos (2 x)}$

Step 2

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

$\frac{lim_{x \to - 8} 9 - lim_{x \to - 8} 8 x + lim_{x \to - 8} \sin (2 x)}{lim_{x \to - 8} 9 x + \cos (2 x)}$

Step 3

Evaluate the limit of $9$ which is constant as $x$ approaches $- 8$ .

$\frac{9 - lim_{x \to - 8} 8 x + lim_{x \to - 8} \sin (2 x)}{lim_{x \to - 8} 9 x + \cos (2 x)}$

Step 4

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{9 - 8 lim_{x \to - 8} x + lim_{x \to - 8} \sin (2 x)}{lim_{x \to - 8} 9 x + \cos (2 x)}$

Step 5

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

$\frac{9 - 8 lim_{x \to - 8} x + \sin (lim_{x \to - 8} 2 x)}{lim_{x \to - 8} 9 x + \cos (2 x)}$

Step 6

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{9 - 8 lim_{x \to - 8} x + \sin (2 lim_{x \to - 8} x)}{lim_{x \to - 8} 9 x + \cos (2 x)}$

Step 7

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

$\frac{9 - 8 lim_{x \to - 8} x + \sin (2 lim_{x \to - 8} x)}{lim_{x \to - 8} 9 x + lim_{x \to - 8} \cos (2 x)}$

Step 8

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\frac{9 - 8 lim_{x \to - 8} x + \sin (2 lim_{x \to - 8} x)}{9 lim_{x \to - 8} x + lim_{x \to - 8} \cos (2 x)}$

Step 9

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

$\frac{9 - 8 lim_{x \to - 8} x + \sin (2 lim_{x \to - 8} x)}{9 lim_{x \to - 8} x + \cos (lim_{x \to - 8} 2 x)}$

Step 10

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{9 - 8 lim_{x \to - 8} x + \sin (2 lim_{x \to - 8} x)}{9 lim_{x \to - 8} x + \cos (2 lim_{x \to - 8} x)}$

Step 11

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

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Step 11.1

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

$\frac{9 - 8 \cdot - 8 + \sin (2 lim_{x \to - 8} x)}{9 lim_{x \to - 8} x + \cos (2 lim_{x \to - 8} x)}$

Step 11.2

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

$\frac{9 - 8 \cdot - 8 + \sin (2 \cdot - 8)}{9 lim_{x \to - 8} x + \cos (2 lim_{x \to - 8} x)}$

Step 11.3

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

$\frac{9 - 8 \cdot - 8 + \sin (2 \cdot - 8)}{9 \cdot - 8 + \cos (2 lim_{x \to - 8} x)}$

Step 11.4

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

$\frac{9 - 8 \cdot - 8 + \sin (2 \cdot - 8)}{9 \cdot - 8 + \cos (2 \cdot - 8)}$

Step 12

Simplify the answer.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

Multiply $- 8$ by $- 8$ .

$\frac{9 + 64 + \sin (2 \cdot - 8)}{9 \cdot - 8 + \cos (2 \cdot - 8)}$

Step 12.1.2

Multiply $2$ by $- 8$ .

$\frac{9 + 64 + \sin (- 16)}{9 \cdot - 8 + \cos (2 \cdot - 8)}$

Step 12.1.3

Evaluate $\sin (- 16)$ .

$\frac{9 + 64 - 0.27563735}{9 \cdot - 8 + \cos (2 \cdot - 8)}$

Step 12.1.4

Add $9$ and $64$ .

$\frac{73 - 0.27563735}{9 \cdot - 8 + \cos (2 \cdot - 8)}$

Step 12.1.5

Subtract $0.27563735$ from $73$ .

$\frac{72.72436264}{9 \cdot - 8 + \cos (2 \cdot - 8)}$

Step 12.2

Simplify the denominator.

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Step 12.2.1

Multiply $9$ by $- 8$ .

$\frac{72.72436264}{- 72 + \cos (2 \cdot - 8)}$

Step 12.2.2

Multiply $2$ by $- 8$ .

$\frac{72.72436264}{- 72 + \cos (- 16)}$

Step 12.2.3

Evaluate $\cos (- 16)$ .

$\frac{72.72436264}{- 72 + 0.96126169}$

Step 12.2.4

Add $- 72$ and $0.96126169$ .

$\frac{72.72436264}{- 71.0387383}$

Step 12.3

Divide $72.72436264$ by $- 71.0387383$ .

$- 1.02372824$