Calculus Examples

$lim_{x \to 8} \frac{\cos (x) + 5}{\cos (x)}$

Step 1

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

$\frac{lim_{x \to 8} \cos (x) + 5}{lim_{x \to 8} \cos (x)}$

Step 2

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

$\frac{lim_{x \to 8} \cos (x) + lim_{x \to 8} 5}{lim_{x \to 8} \cos (x)}$

Step 3

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

$\frac{\cos (lim_{x \to 8} x) + lim_{x \to 8} 5}{lim_{x \to 8} \cos (x)}$

Step 4

Evaluate the limit of $5$ which is constant as $x$ approaches $8$ .

$\frac{\cos (lim_{x \to 8} x) + 5}{lim_{x \to 8} \cos (x)}$

Step 5

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

$\frac{\cos (lim_{x \to 8} x) + 5}{\cos (lim_{x \to 8} x)}$

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 $x$ .

$\frac{\cos (8) + 5}{\cos (lim_{x \to 8} x)}$

Step 6.2

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

$\frac{\cos (8) + 5}{\cos (8)}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Evaluate $\cos (8)$ .

$\frac{0.99026806 + 5}{\cos (8)}$

Step 7.1.2

Add $0.99026806$ and $5$ .

$\frac{5.99026806}{\cos (8)}$

Step 7.2

Evaluate $\cos (8)$ .

$\frac{5.99026806}{0.99026806}$

Step 7.3

Divide $5.99026806$ by $0.99026806$ .

$6.04913786$