Calculus Examples

lim_{x \to 5} \csc (\frac{π x}{4})

$lim_{x \to 5} \csc (\frac{π x}{4})$

Step 1

Evaluate the limit.

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Step 1.1

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

\csc (lim_{x \to 5} \frac{π x}{4})

$\csc (lim_{x \to 5} \frac{π x}{4})$

Step 1.2

Move the term

\frac{π}{4}

$\frac{π}{4}$ outside of the limit because it is constant with respect to

x

$x$ .

\csc (\frac{π}{4} lim_{x \to 5} x)

$\csc (\frac{π}{4} lim_{x \to 5} x)$

\csc (\frac{π}{4} lim_{x \to 5} x)

$\csc (\frac{π}{4} lim_{x \to 5} x)$

Step 2

Evaluate the limit of

x

$x$ by plugging in

5

$5$ for

x

$x$ .

\csc (\frac{π}{4} \cdot 5)

$\csc (\frac{π}{4} \cdot 5)$

Step 3

Simplify the answer.

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Step 3.1

Combine

\frac{π}{4}

$\frac{π}{4}$ and

5

$5$ .

\csc (\frac{π \cdot 5}{4})

$\csc (\frac{π \cdot 5}{4})$

Step 3.2

Move

5

$5$ to the left of

π

$π$ .

\csc (\frac{5 \cdot π}{4})

$\csc (\frac{5 \cdot π}{4})$

Step 3.3

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

- \csc (\frac{π}{4})

$- \csc (\frac{π}{4})$

Step 3.4

The exact value of

\csc (\frac{π}{4})

$\csc (\frac{π}{4})$ is

\sqrt{2}

$\sqrt{2}$ .

- \sqrt{2}

$- \sqrt{2}$

- \sqrt{2}

$- \sqrt{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

- \sqrt{2}

$- \sqrt{2}$

Decimal Form:

- 1.41421356 \dots

$- 1.41421356 \dots$