Calculus Examples

$lim_{x \to \frac{π}{3}} \cot (- \frac{2 \sqrt{3} x}{π} + \csc (x) - \frac{π}{4})$

Step 1

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

$\cot (lim_{x \to \frac{π}{3}} - \frac{2 \sqrt{3} x}{π} + \csc (x) - \frac{π}{4})$

Step 2

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

$\cot (- lim_{x \to \frac{π}{3}} \frac{2 \sqrt{3} x}{π} + lim_{x \to \frac{π}{3}} \csc (x) - lim_{x \to \frac{π}{3}} \frac{π}{4})$

Step 3

Move the term $\frac{2 \sqrt{3}}{π}$ outside of the limit because it is constant with respect to $x$ .

$\cot (- (\frac{2 \sqrt{3}}{π} lim_{x \to \frac{π}{3}} x) + lim_{x \to \frac{π}{3}} \csc (x) - lim_{x \to \frac{π}{3}} \frac{π}{4})$

Step 4

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

$\cot (- (\frac{2 \sqrt{3}}{π} lim_{x \to \frac{π}{3}} x) + \csc (lim_{x \to \frac{π}{3}} x) - lim_{x \to \frac{π}{3}} \frac{π}{4})$

Step 5

Evaluate the limit of $\frac{π}{4}$ which is constant as $x$ approaches $\frac{π}{3}$ .

$\cot (- (\frac{2 \sqrt{3}}{π} lim_{x \to \frac{π}{3}} x) + \csc (lim_{x \to \frac{π}{3}} x) - \frac{π}{4})$

Step 6

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

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Step 6.1

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

$\cot (- (\frac{2 \sqrt{3}}{π} \cdot \frac{π}{3}) + \csc (lim_{x \to \frac{π}{3}} x) - \frac{π}{4})$

Step 6.2

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

$\cot (- (\frac{2 \sqrt{3}}{π} \cdot \frac{π}{3}) + \csc (\frac{π}{3}) - \frac{π}{4})$

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

Cancel the common factor of $π$ .

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Step 7.1.1.1

Cancel the common factor.

$\cot (- (\frac{2 \sqrt{3}}{π} \cdot \frac{π}{3}) + \csc (\frac{π}{3}) - \frac{π}{4})$

Step 7.1.1.2

Rewrite the expression.

$\cot (- (2 \sqrt{3} \frac{1}{3}) + \csc (\frac{π}{3}) - \frac{π}{4})$

Step 7.1.2

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

$\cot (- (\frac{2}{3} \sqrt{3}) + \csc (\frac{π}{3}) - \frac{π}{4})$

Step 7.1.3

Combine $\frac{2}{3}$ and $\sqrt{3}$ .

$\cot (- \frac{2 \sqrt{3}}{3} + \csc (\frac{π}{3}) - \frac{π}{4})$

Step 7.1.4

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

$\cot (- \frac{2 \sqrt{3}}{3} + \frac{2}{\sqrt{3}} - \frac{π}{4})$

Step 7.1.5

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\cot (- \frac{2 \sqrt{3}}{3} + \frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} - \frac{π}{4})$

Step 7.1.6

Combine and simplify the denominator.

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Step 7.1.6.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\cot (- \frac{2 \sqrt{3}}{3} + \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} - \frac{π}{4})$

Step 7.1.6.2

Raise $\sqrt{3}$ to the power of $1$ .

$\cot (- \frac{2 \sqrt{3}}{3} + \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} - \frac{π}{4})$

Step 7.1.6.3

Raise $\sqrt{3}$ to the power of $1$ .

$\cot (- \frac{2 \sqrt{3}}{3} + \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} - \frac{π}{4})$

Step 7.1.6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\cot (- \frac{2 \sqrt{3}}{3} + \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} - \frac{π}{4})$

Step 7.1.6.5

Add $1$ and $1$ .

$\cot (- \frac{2 \sqrt{3}}{3} + \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}} - \frac{π}{4})$

Step 7.1.6.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 7.1.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\cot (- \frac{2 \sqrt{3}}{3} + \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} - \frac{π}{4})$

Step 7.1.6.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\cot (- \frac{2 \sqrt{3}}{3} + \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} - \frac{π}{4})$

Step 7.1.6.6.3

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

$\cot (- \frac{2 \sqrt{3}}{3} + \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} - \frac{π}{4})$

Step 7.1.6.6.4

Cancel the common factor of $2$ .

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Step 7.1.6.6.4.1

Cancel the common factor.

$\cot (- \frac{2 \sqrt{3}}{3} + \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} - \frac{π}{4})$

Step 7.1.6.6.4.2

Rewrite the expression.

$\cot (- \frac{2 \sqrt{3}}{3} + \frac{2 \sqrt{3}}{3^{1}} - \frac{π}{4})$

Step 7.1.6.6.5

Evaluate the exponent.

$\cot (- \frac{2 \sqrt{3}}{3} + \frac{2 \sqrt{3}}{3} - \frac{π}{4})$

Step 7.2

Combine the numerators over the common denominator.

$\cot (\frac{- 2 \sqrt{3} + 2 \sqrt{3}}{3} + \frac{- π}{4})$

Step 7.3

Add $- 2 \sqrt{3}$ and $2 \sqrt{3}$ .

$\cot (\frac{0}{3} + \frac{- π}{4})$

Step 7.4

Simplify each term.

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Step 7.4.1

Divide $0$ by $3$ .

$\cot (0 + \frac{- π}{4})$

Step 7.4.2

Move the negative in front of the fraction.

$\cot (0 - \frac{π}{4})$

Step 7.5

Subtract $\frac{π}{4}$ from $0$ .

$\cot (- \frac{π}{4})$

Step 7.6

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\cot (\frac{7 π}{4})$

Step 7.7

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cotangent is negative in the fourth quadrant.

$- \cot (\frac{π}{4})$

Step 7.8

The exact value of $\cot (\frac{π}{4})$ is $1$ .

$- 1 \cdot 1$

Step 7.9

Multiply $- 1$ by $1$ .

$- 1$