Calculus Examples

$lim_{x \to 8} \frac{\sin (x)}{1 - \cos (x)}$

Step 1

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

$\frac{lim_{x \to 8} \sin (x)}{lim_{x \to 8} 1 - \cos (x)}$

Step 2

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

$\frac{\sin (lim_{x \to 8} x)}{lim_{x \to 8} 1 - \cos (x)}$

Step 3

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

$\frac{\sin (lim_{x \to 8} x)}{lim_{x \to 8} 1 - lim_{x \to 8} \cos (x)}$

Step 4

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$\frac{\sin (lim_{x \to 8} x)}{1 - lim_{x \to 8} \cos (x)}$

Step 5

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

$\frac{\sin (lim_{x \to 8} x)}{1 - \cos (lim_{x \to 8} x)}$

Step 6

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

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Step 6.1

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

$\frac{\sin (8)}{1 - \cos (lim_{x \to 8} x)}$

Step 6.2

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

$\frac{\sin (8)}{1 - \cos (8)}$

Step 7

Simplify the answer.

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Step 7.1

Evaluate $\sin (8)$ .

$\frac{0.1391731}{1 - \cos (8)}$

Step 7.2

Simplify the denominator.

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Step 7.2.1

Evaluate $\cos (8)$ .

$\frac{0.1391731}{1 - 1 \cdot 0.99026806}$

Step 7.2.2

Multiply $- 1$ by $0.99026806$ .

$\frac{0.1391731}{1 - 0.99026806}$

Step 7.2.3

Subtract $0.99026806$ from $1$ .

$\frac{0.1391731}{0.00973193}$

Step 7.3

Divide $0.1391731$ by $0.00973193$ .

$14.30066625$