Calculus Examples

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

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

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})

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

Step 2

Split the limit using the Sum of Limits Rule on the limit as

x

$x$ approaches

\frac{π}{3}

$\frac{π}{3}$ .

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})

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

Step 3

Move the term

\frac{2 \sqrt{3}}{π}

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

x

$x$ .

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})

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

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})

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

Step 5

Evaluate the limit of

\frac{π}{4}

$\frac{π}{4}$ which is constant as

x

$x$ approaches

\frac{π}{3}

$\frac{π}{3}$ .

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

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

Step 6

Evaluate the limits by plugging in

\frac{π}{3}

$\frac{π}{3}$ for all occurrences of

x

$x$ .

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Step 6.1

Evaluate the limit of

x

$x$ by plugging in

\frac{π}{3}

$\frac{π}{3}$ for

x

$x$ .

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

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

Step 6.2

Evaluate the limit of

x

$x$ by plugging in

\frac{π}{3}

$\frac{π}{3}$ for

x

$x$ .

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

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

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

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

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

$π$ .

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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

$1$ .

$- 1$