Calculus Examples

$\frac{lim_{x \to 0} \sin (b x) - 4 \sin (a x)}{b x - a x}$

Step 1

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

$\frac{lim_{x \to 0} \sin (b x) - lim_{x \to 0} 4 \sin (a x)}{b x - a x}$

Step 2

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

$\frac{\sin (lim_{x \to 0} b x) - lim_{x \to 0} 4 \sin (a x)}{b x - a x}$

Step 3

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

$\frac{\sin (b lim_{x \to 0} x) - lim_{x \to 0} 4 \sin (a x)}{b x - a x}$

Step 4

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

$\frac{\sin (b lim_{x \to 0} x) - 4 lim_{x \to 0} \sin (a x)}{b x - a x}$

Step 5

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

$\frac{\sin (b lim_{x \to 0} x) - 4 \sin (lim_{x \to 0} a x)}{b x - a x}$

Step 6

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

$\frac{\sin (b lim_{x \to 0} x) - 4 \sin (a lim_{x \to 0} x)}{b x - a x}$

Step 7

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

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Step 7.1

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

$\frac{\sin (b \cdot 0) - 4 \sin (a lim_{x \to 0} x)}{b x - a x}$

Step 7.2

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

$\frac{\sin (b \cdot 0) - 4 \sin (a \cdot 0)}{b x - a x}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Multiply $b$ by $0$ .

$\frac{\sin (0) - 4 \sin (a \cdot 0)}{b x - a x}$

Step 8.1.2

The exact value of $\sin (0)$ is $0$ .

$\frac{0 - 4 \sin (a \cdot 0)}{b x - a x}$

Step 8.1.3

Multiply $a$ by $0$ .

$\frac{0 - 4 \sin (0)}{b x - a x}$

Step 8.1.4

The exact value of $\sin (0)$ is $0$ .

$\frac{0 - 4 \cdot 0}{b x - a x}$

Step 8.1.5

Multiply $- 4$ by $0$ .

$\frac{0 + 0}{b x - a x}$

Step 8.1.6

Add $0$ and $0$ .

$\frac{0}{b x - a x}$

Step 8.2

Factor $x$ out of $b x - a x$ .

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Step 8.2.1

Factor $x$ out of $b x$ .

$\frac{0}{x b - a x}$

Step 8.2.2

Factor $x$ out of $- a x$ .

$\frac{0}{x b + x (- a)}$

Step 8.2.3

Factor $x$ out of $x b + x (- a)$ .

$\frac{0}{x (b - a)}$

Step 8.3

Divide $0$ by $x (b - a)$ .

$0$