Calculus Examples

$lim_{x \to \frac{π}{6}} \frac{2 \sin (x) + \sin (x) - 1}{2 \sin (x) - 3 \sin (x) + 1}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\frac{π}{6}$ .

$\frac{lim_{x \to \frac{π}{6}} 2 \sin (x) + \sin (x) - 1}{lim_{x \to \frac{π}{6}} 2 \sin (x) - 3 \sin (x) + 1}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{π}{6}$ .

$\frac{lim_{x \to \frac{π}{6}} 2 \sin (x) + lim_{x \to \frac{π}{6}} \sin (x) - lim_{x \to \frac{π}{6}} 1}{lim_{x \to \frac{π}{6}} 2 \sin (x) - 3 \sin (x) + 1}$

Step 3

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 lim_{x \to \frac{π}{6}} \sin (x) + lim_{x \to \frac{π}{6}} \sin (x) - lim_{x \to \frac{π}{6}} 1}{lim_{x \to \frac{π}{6}} 2 \sin (x) - 3 \sin (x) + 1}$

Step 4

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

$\frac{2 \sin (lim_{x \to \frac{π}{6}} x) + lim_{x \to \frac{π}{6}} \sin (x) - lim_{x \to \frac{π}{6}} 1}{lim_{x \to \frac{π}{6}} 2 \sin (x) - 3 \sin (x) + 1}$

Step 5

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

$\frac{2 \sin (lim_{x \to \frac{π}{6}} x) + \sin (lim_{x \to \frac{π}{6}} x) - lim_{x \to \frac{π}{6}} 1}{lim_{x \to \frac{π}{6}} 2 \sin (x) - 3 \sin (x) + 1}$

Step 6

Evaluate the limit of $1$ which is constant as $x$ approaches $\frac{π}{6}$ .

$\frac{2 \sin (lim_{x \to \frac{π}{6}} x) + \sin (lim_{x \to \frac{π}{6}} x) - 1 \cdot 1}{lim_{x \to \frac{π}{6}} 2 \sin (x) - 3 \sin (x) + 1}$

Step 7

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{π}{6}$ .

$\frac{2 \sin (lim_{x \to \frac{π}{6}} x) + \sin (lim_{x \to \frac{π}{6}} x) - 1 \cdot 1}{lim_{x \to \frac{π}{6}} 2 \sin (x) - lim_{x \to \frac{π}{6}} 3 \sin (x) + lim_{x \to \frac{π}{6}} 1}$

Step 8

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 \sin (lim_{x \to \frac{π}{6}} x) + \sin (lim_{x \to \frac{π}{6}} x) - 1 \cdot 1}{2 lim_{x \to \frac{π}{6}} \sin (x) - lim_{x \to \frac{π}{6}} 3 \sin (x) + lim_{x \to \frac{π}{6}} 1}$

Step 9

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

$\frac{2 \sin (lim_{x \to \frac{π}{6}} x) + \sin (lim_{x \to \frac{π}{6}} x) - 1 \cdot 1}{2 \sin (lim_{x \to \frac{π}{6}} x) - lim_{x \to \frac{π}{6}} 3 \sin (x) + lim_{x \to \frac{π}{6}} 1}$

Step 10

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 \sin (lim_{x \to \frac{π}{6}} x) + \sin (lim_{x \to \frac{π}{6}} x) - 1 \cdot 1}{2 \sin (lim_{x \to \frac{π}{6}} x) - 3 lim_{x \to \frac{π}{6}} \sin (x) + lim_{x \to \frac{π}{6}} 1}$

Step 11

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

$\frac{2 \sin (lim_{x \to \frac{π}{6}} x) + \sin (lim_{x \to \frac{π}{6}} x) - 1 \cdot 1}{2 \sin (lim_{x \to \frac{π}{6}} x) - 3 \sin (lim_{x \to \frac{π}{6}} x) + lim_{x \to \frac{π}{6}} 1}$

Step 12

Evaluate the limit of $1$ which is constant as $x$ approaches $\frac{π}{6}$ .

$\frac{2 \sin (lim_{x \to \frac{π}{6}} x) + \sin (lim_{x \to \frac{π}{6}} x) - 1 \cdot 1}{2 \sin (lim_{x \to \frac{π}{6}} x) - 3 \sin (lim_{x \to \frac{π}{6}} x) + 1}$

Step 13

Evaluate the limits by plugging in $\frac{π}{6}$ for all occurrences of $x$ .

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Step 13.1

Evaluate the limit of $x$ by plugging in $\frac{π}{6}$ for $x$ .

$\frac{2 \sin (\frac{π}{6}) + \sin (lim_{x \to \frac{π}{6}} x) - 1 \cdot 1}{2 \sin (lim_{x \to \frac{π}{6}} x) - 3 \sin (lim_{x \to \frac{π}{6}} x) + 1}$

Step 13.2

Evaluate the limit of $x$ by plugging in $\frac{π}{6}$ for $x$ .

$\frac{2 \sin (\frac{π}{6}) + \sin (\frac{π}{6}) - 1 \cdot 1}{2 \sin (lim_{x \to \frac{π}{6}} x) - 3 \sin (lim_{x \to \frac{π}{6}} x) + 1}$

Step 13.3

Evaluate the limit of $x$ by plugging in $\frac{π}{6}$ for $x$ .

$\frac{2 \sin (\frac{π}{6}) + \sin (\frac{π}{6}) - 1 \cdot 1}{2 \sin (\frac{π}{6}) - 3 \sin (lim_{x \to \frac{π}{6}} x) + 1}$

Step 13.4

Evaluate the limit of $x$ by plugging in $\frac{π}{6}$ for $x$ .

$\frac{2 \sin (\frac{π}{6}) + \sin (\frac{π}{6}) - 1 \cdot 1}{2 \sin (\frac{π}{6}) - 3 \sin (\frac{π}{6}) + 1}$

Step 14

Simplify the answer.

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Step 14.1

Simplify the numerator.

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Step 14.1.1

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$\frac{2 (\frac{1}{2}) + \sin (\frac{π}{6}) - 1 \cdot 1}{2 \sin (\frac{π}{6}) - 3 \sin (\frac{π}{6}) + 1}$

Step 14.1.2

Cancel the common factor of $2$ .

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Step 14.1.2.1

Cancel the common factor.

$\frac{2 (\frac{1}{2}) + \sin (\frac{π}{6}) - 1 \cdot 1}{2 \sin (\frac{π}{6}) - 3 \sin (\frac{π}{6}) + 1}$

Step 14.1.2.2

Rewrite the expression.

$\frac{1 + \sin (\frac{π}{6}) - 1 \cdot 1}{2 \sin (\frac{π}{6}) - 3 \sin (\frac{π}{6}) + 1}$

Step 14.1.3

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$\frac{1 + \frac{1}{2} - 1 \cdot 1}{2 \sin (\frac{π}{6}) - 3 \sin (\frac{π}{6}) + 1}$

Step 14.1.4

Multiply $- 1$ by $1$ .

$\frac{1 + \frac{1}{2} - 1}{2 \sin (\frac{π}{6}) - 3 \sin (\frac{π}{6}) + 1}$

Step 14.1.5

Write $1$ as a fraction with a common denominator.

$\frac{\frac{2}{2} + \frac{1}{2} - 1}{2 \sin (\frac{π}{6}) - 3 \sin (\frac{π}{6}) + 1}$

Step 14.1.6

Combine the numerators over the common denominator.

$\frac{\frac{2 + 1}{2} - 1}{2 \sin (\frac{π}{6}) - 3 \sin (\frac{π}{6}) + 1}$

Step 14.1.7

Add $2$ and $1$ .

$\frac{\frac{3}{2} - 1}{2 \sin (\frac{π}{6}) - 3 \sin (\frac{π}{6}) + 1}$

Step 14.1.8

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\frac{3}{2} - 1 \cdot \frac{2}{2}}{2 \sin (\frac{π}{6}) - 3 \sin (\frac{π}{6}) + 1}$

Step 14.1.9

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}{2 \sin (\frac{π}{6}) - 3 \sin (\frac{π}{6}) + 1}$

Step 14.1.10

Combine the numerators over the common denominator.

$\frac{\frac{3 - 1 \cdot 2}{2}}{2 \sin (\frac{π}{6}) - 3 \sin (\frac{π}{6}) + 1}$

Step 14.1.11

Simplify the numerator.

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Step 14.1.11.1

Multiply $- 1$ by $2$ .

$\frac{\frac{3 - 2}{2}}{2 \sin (\frac{π}{6}) - 3 \sin (\frac{π}{6}) + 1}$

Step 14.1.11.2

Subtract $2$ from $3$ .

$\frac{\frac{1}{2}}{2 \sin (\frac{π}{6}) - 3 \sin (\frac{π}{6}) + 1}$

Step 14.2

Simplify the denominator.

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Step 14.2.1

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$\frac{\frac{1}{2}}{2 (\frac{1}{2}) - 3 \sin (\frac{π}{6}) + 1}$

Step 14.2.2

Cancel the common factor of $2$ .

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Step 14.2.2.1

Cancel the common factor.

$\frac{\frac{1}{2}}{2 (\frac{1}{2}) - 3 \sin (\frac{π}{6}) + 1}$

Step 14.2.2.2

Rewrite the expression.

$\frac{\frac{1}{2}}{1 - 3 \sin (\frac{π}{6}) + 1}$

Step 14.2.3

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$\frac{\frac{1}{2}}{1 - 3 (\frac{1}{2}) + 1}$

Step 14.2.4

Combine $- 3$ and $\frac{1}{2}$ .

$\frac{\frac{1}{2}}{1 + \frac{- 3}{2} + 1}$

Step 14.2.5

Move the negative in front of the fraction.

$\frac{\frac{1}{2}}{1 - \frac{3}{2} + 1}$

Step 14.2.6

Write $1$ as a fraction with a common denominator.

$\frac{\frac{1}{2}}{\frac{2}{2} - \frac{3}{2} + 1}$

Step 14.2.7

Combine the numerators over the common denominator.

$\frac{\frac{1}{2}}{\frac{2 - 3}{2} + 1}$

Step 14.2.8

Subtract $3$ from $2$ .

$\frac{\frac{1}{2}}{\frac{- 1}{2} + 1}$

Step 14.2.9

Write $1$ as a fraction with a common denominator.

$\frac{\frac{1}{2}}{\frac{- 1}{2} + \frac{2}{2}}$

Step 14.2.10

Combine the numerators over the common denominator.

$\frac{\frac{1}{2}}{\frac{- 1 + 2}{2}}$

Step 14.2.11

Add $- 1$ and $2$ .

$\frac{\frac{1}{2}}{\frac{1}{2}}$

Step 14.3

Cancel the common factor of $\frac{1}{2}$ .

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Step 14.3.1

Cancel the common factor.

$\frac{\frac{1}{2}}{\frac{1}{2}}$

Step 14.3.2

Rewrite the expression.

$1$