Calculus Examples

$lim_{x \to 5} \frac{\sin (6 x)}{x - 5 π}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $5$ .

$\frac{lim_{x \to 5} \sin (6 x)}{lim_{x \to 5} x - 5 π}$

Step 2

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

$\frac{\sin (lim_{x \to 5} 6 x)}{lim_{x \to 5} x - 5 π}$

Step 3

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sin (6 lim_{x \to 5} x)}{lim_{x \to 5} x - 5 π}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $5$ .

$\frac{\sin (6 lim_{x \to 5} x)}{lim_{x \to 5} x - lim_{x \to 5} 5 π}$

Step 5

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

$\frac{\sin (6 lim_{x \to 5} x)}{lim_{x \to 5} x - 5 π}$

Step 6

Evaluate the limits by plugging in $5$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{\sin (6 \cdot 5)}{lim_{x \to 5} x - 5 π}$

Step 6.2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{\sin (6 \cdot 5)}{5 - 5 π}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Multiply $6$ by $5$ .

$\frac{\sin (30)}{5 - 5 π}$

Step 7.1.2

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$\frac{\frac{1}{2}}{5 - 5 π}$

Step 7.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{2} \cdot \frac{1}{5 - 5 π}$

Step 7.3

Multiply $\frac{1}{2}$ by $\frac{1}{5 - 5 π}$ .

$\frac{1}{2 (5 - 5 π)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2 (5 - 5 π)}$

Decimal Form:

$- 0.04669422 \dots$