Calculus Examples

$\frac{lim_{a \to \frac{π}{3}} \cos (a) - (\frac{1}{2})}{a - \frac{π}{3}}$

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $a$ approaches $\frac{π}{3}$ .

$\frac{lim_{a \to \frac{π}{3}} \cos (a) - lim_{a \to \frac{π}{3}} \frac{1}{2}}{a - \frac{π}{3}}$

Step 1.2

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

$\frac{\cos (lim_{a \to \frac{π}{3}} a) - lim_{a \to \frac{π}{3}} \frac{1}{2}}{a - \frac{π}{3}}$

Step 1.3

Evaluate the limit of $\frac{1}{2}$ which is constant as $a$ approaches $\frac{π}{3}$ .

$\frac{\cos (lim_{a \to \frac{π}{3}} a) - \frac{1}{2}}{a - \frac{π}{3}}$

Step 2

Evaluate the limit of $a$ by plugging in $\frac{π}{3}$ for $a$ .

$\frac{\cos (\frac{π}{3}) - \frac{1}{2}}{a - \frac{π}{3}}$

Step 3

Simplify the answer.

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Step 3.1

Multiply the numerator and denominator of the fraction by $6$ .

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Step 3.1.1

Multiply $\frac{\cos (\frac{π}{3}) - \frac{1}{2}}{a - \frac{π}{3}}$ by $\frac{6}{6}$ .

$\frac{6}{6} \cdot \frac{\cos (\frac{π}{3}) - \frac{1}{2}}{a - \frac{π}{3}}$

Step 3.1.2

Combine.

$\frac{6 (\cos (\frac{π}{3}) - \frac{1}{2})}{6 (a - \frac{π}{3})}$

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $2$ .

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Step 3.3.1.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 3.3.1.2

Factor $2$ out of $6$ .

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

Step 3.3.1.3

Cancel the common factor.

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

Step 3.3.1.4

Rewrite the expression.

$\frac{6 \cos (\frac{π}{3}) + 3 \cdot - 1}{6 a + 6 (- \frac{π}{3})}$

Step 3.3.2

Multiply $3$ by $- 1$ .

$\frac{6 \cos (\frac{π}{3}) - 3}{6 a + 6 (- \frac{π}{3})}$

Step 3.3.3

Cancel the common factor of $3$ .

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Step 3.3.3.1

Move the leading negative in $- \frac{π}{3}$ into the numerator.

$\frac{6 \cos (\frac{π}{3}) - 3}{6 a + 6 \frac{- π}{3}}$

Step 3.3.3.2

Factor $3$ out of $6$ .

$\frac{6 \cos (\frac{π}{3}) - 3}{6 a + 3 (2) \frac{- π}{3}}$

Step 3.3.3.3

Cancel the common factor.

$\frac{6 \cos (\frac{π}{3}) - 3}{6 a + 3 \cdot 2 \frac{- π}{3}}$

Step 3.3.3.4

Rewrite the expression.

$\frac{6 \cos (\frac{π}{3}) - 3}{6 a + 2 (- π)}$

Step 3.3.4

Multiply $- 1$ by $2$ .

$\frac{6 \cos (\frac{π}{3}) - 3}{6 a - 2 π}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

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

Step 3.4.2

Cancel the common factor of $2$ .

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Step 3.4.2.1

Factor $2$ out of $6$ .

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

Step 3.4.2.2

Cancel the common factor.

$\frac{2 \cdot 3 \frac{1}{2} - 3}{6 a - 2 π}$

Step 3.4.2.3

Rewrite the expression.

$\frac{3 - 3}{6 a - 2 π}$

Step 3.4.3

Subtract $3$ from $3$ .

$\frac{0}{6 a - 2 π}$

Step 3.5

Factor $2$ out of $6 a - 2 π$ .

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Step 3.5.1

Factor $2$ out of $6 a$ .

$\frac{0}{2 (3 a) - 2 π}$

Step 3.5.2

Factor $2$ out of $- 2 π$ .

$\frac{0}{2 (3 a) + 2 (- π)}$

Step 3.5.3

Factor $2$ out of $2 (3 a) + 2 (- π)$ .

$\frac{0}{2 (3 a - π)}$

Step 3.6

Cancel the common factor of $0$ and $2$ .

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Step 3.6.1

Factor $2$ out of $0$ .

$\frac{2 (0)}{2 (3 a - π)}$

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Cancel the common factor.

$\frac{2 \cdot 0}{2 (3 a - π)}$

Step 3.6.2.2

Rewrite the expression.

$\frac{0}{3 a - π}$

Step 3.7

Divide $0$ by $3 a - π$ .

$0$