Calculus Examples

$lim_{x \to 0} \frac{\sin (5 x + 10)}{4 x + 8}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} \sin (5 x + 10)}{lim_{x \to 0} 4 x + 8}$

Step 2

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

$\frac{\sin (lim_{x \to 0} 5 x + 10)}{lim_{x \to 0} 4 x + 8}$

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\sin (lim_{x \to 0} 5 x + lim_{x \to 0} 10)}{lim_{x \to 0} 4 x + 8}$

Step 4

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

$\frac{\sin (5 lim_{x \to 0} x + lim_{x \to 0} 10)}{lim_{x \to 0} 4 x + 8}$

Step 5

Evaluate the limit of $10$ which is constant as $x$ approaches $0$ .

$\frac{\sin (5 lim_{x \to 0} x + 10)}{lim_{x \to 0} 4 x + 8}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\sin (5 lim_{x \to 0} x + 10)}{lim_{x \to 0} 4 x + lim_{x \to 0} 8}$

Step 7

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sin (5 lim_{x \to 0} x + 10)}{4 lim_{x \to 0} x + lim_{x \to 0} 8}$

Step 8

Evaluate the limit of $8$ which is constant as $x$ approaches $0$ .

$\frac{\sin (5 lim_{x \to 0} x + 10)}{4 lim_{x \to 0} x + 8}$

Step 9

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 9.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\sin (5 \cdot 0 + 10)}{4 lim_{x \to 0} x + 8}$

Step 9.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\sin (5 \cdot 0 + 10)}{4 \cdot 0 + 8}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Multiply $5$ by $0$ .

$\frac{\sin (0 + 10)}{4 \cdot 0 + 8}$

Step 10.1.2

Add $0$ and $10$ .

$\frac{\sin (10)}{4 \cdot 0 + 8}$

Step 10.1.3

Evaluate $\sin (10)$ .

$\frac{0.17364817}{4 \cdot 0 + 8}$

Step 10.2

Simplify the denominator.

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Step 10.2.1

Multiply $4$ by $0$ .

$\frac{0.17364817}{0 + 8}$

Step 10.2.2

Add $0$ and $8$ .

$\frac{0.17364817}{8}$

Step 10.3

Divide $0.17364817$ by $8$ .

$0.02170602$