Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Move the exponent $2$ from $\cos^{2} (x)$ outside the limit using the Limits Power Rule.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Evaluate the limits by plugging in $\frac{π}{3}$ for all occurrences of $x$ .

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Step 8.1

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

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

Step 8.2

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

$\frac{2 \cos^{2} (\frac{π}{3}) + 3 \cos (\frac{π}{3}) - 1 \cdot 2}{2 \cos (x) - 1}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Factor by grouping.

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Step 9.1.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 2 = - 4$ and whose sum is $b = 3$ .

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Step 9.1.1.1.1

Multiply $- 1$ by $2$ .

$\frac{2 \cos^{2} (\frac{π}{3}) + 3 \cos (\frac{π}{3}) - 2}{2 \cos (x) - 1}$

Step 9.1.1.1.2

Factor $3$ out of $3 \cos (\frac{π}{3})$ .

$\frac{2 \cos^{2} (\frac{π}{3}) + 3 (\cos (\frac{π}{3})) - 2}{2 \cos (x) - 1}$

Step 9.1.1.1.3

Rewrite $3$ as $- 1$ plus $4$

$\frac{2 \cos^{2} (\frac{π}{3}) + (- 1 + 4) \cos (\frac{π}{3}) - 2}{2 \cos (x) - 1}$

Step 9.1.1.1.4

Apply the distributive property.

$\frac{2 \cos^{2} (\frac{π}{3}) - 1 \cos (\frac{π}{3}) + 4 \cos (\frac{π}{3}) - 2}{2 \cos (x) - 1}$

Step 9.1.1.2

Factor out the greatest common factor from each group.

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Step 9.1.1.2.1

Group the first two terms and the last two terms.

$\frac{(2 \cos^{2} (\frac{π}{3}) - 1 \cos (\frac{π}{3})) + 4 \cos (\frac{π}{3}) - 2}{2 \cos (x) - 1}$

Step 9.1.1.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 9.1.1.3

Factor the polynomial by factoring out the greatest common factor, $2 \cos (\frac{π}{3}) - 1$ .

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

Step 9.1.2

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

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

Step 9.1.3

Cancel the common factor of $2$ .

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Step 9.1.3.1

Cancel the common factor.

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

Step 9.1.3.2

Rewrite the expression.

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

Step 9.1.4

Subtract $1$ from $1$ .

$\frac{0 (\cos (\frac{π}{3}) + 2)}{2 \cos (x) - 1}$

Step 9.1.5

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

$\frac{0 (\frac{1}{2} + 2)}{2 \cos (x) - 1}$

Step 9.1.6

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

$\frac{0 (\frac{1}{2} + 2 \cdot \frac{2}{2})}{2 \cos (x) - 1}$

Step 9.1.7

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

$\frac{0 (\frac{1}{2} + \frac{2 \cdot 2}{2})}{2 \cos (x) - 1}$

Step 9.1.8

Combine the numerators over the common denominator.

$\frac{0 \frac{1 + 2 \cdot 2}{2}}{2 \cos (x) - 1}$

Step 9.1.9

Simplify the numerator.

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Step 9.1.9.1

Multiply $2$ by $2$ .

$\frac{0 \frac{1 + 4}{2}}{2 \cos (x) - 1}$

Step 9.1.9.2

Add $1$ and $4$ .

$\frac{0 (\frac{5}{2})}{2 \cos (x) - 1}$

Step 9.2

Multiply $0$ by $\frac{5}{2}$ .

$\frac{0}{2 \cos (x) - 1}$

Step 9.3

Divide $0$ by $2 \cos (x) - 1$ .

$0$