Calculus Examples

$lim_{x \to \frac{5 π}{2^{-}}} x^{2} \cot (x)$

Step 1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $\frac{5 π}{2^{-}}$ .

$lim_{x \to \frac{5 π}{2^{-}}} x^{2} \cdot lim_{x \to \frac{5 π}{2^{-}}} \cot (x)$

Step 2

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

${(lim_{x \to \frac{5 π}{2^{-}}} x)}^{2} \cdot lim_{x \to \frac{5 π}{2^{-}}} \cot (x)$

Step 3

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

${(lim_{x \to \frac{5 π}{2^{-}}} x)}^{2} \cdot \cot (lim_{x \to \frac{5 π}{2^{-}}} x)$

Step 4

Evaluate the limits by plugging in $\frac{5 π}{2^{-}}$ for all occurrences of $x$ .

Tap for more steps...

Step 4.1

Evaluate the limit of $x$ by plugging in $\frac{5 π}{2^{-}}$ for $x$ .

${(\frac{5 π}{2^{-}})}^{2} \cdot \cot (lim_{x \to \frac{5 π}{2^{-}}} x)$

Step 4.2

Evaluate the limit of $x$ by plugging in $\frac{5 π}{2^{-}}$ for $x$ .

${(\frac{5 π}{2^{-}})}^{2} \cdot \cot (\frac{5 π}{2^{-}})$

Step 5

Simplify the answer.

Tap for more steps...

Step 5.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 5.1.1

Apply the product rule to $\frac{5 π}{2^{-}}$ .

$\frac{{(5 π)}^{2}}{{(2^{-})}^{2}} \cdot \cot (\frac{5 π}{2^{-}})$

Step 5.1.2

Apply the product rule to $5 π$ .

$\frac{5^{2} π^{2}}{{(2^{-})}^{2}} \cdot \cot (\frac{5 π}{2^{-}})$

Step 5.2

Raise $5$ to the power of $2$ .

$\frac{25 π^{2}}{{(2^{-})}^{2}} \cdot \cot (\frac{5 π}{2^{-}})$

Step 5.3

Multiply the exponents in ${(2^{-})}^{2}$ .

Tap for more steps...

Step 5.3.1

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

$\frac{25 π^{2}}{2^{- \cdot 2}} \cdot \cot (\frac{5 π}{2^{-}})$

Step 5.3.2

Move $2$ to the left of $-$ .

$\frac{25 π^{2}}{2^{2 -}} \cdot \cot (\frac{5 π}{2^{-}})$

Step 5.4

Combine $\frac{25 π^{2}}{2^{2 -}}$ and $\cot (\frac{5 π}{2^{-}})$ .

$\frac{25 π^{2} \cot (\frac{5 π}{2^{-}})}{2^{2 -}}$