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1

Example

$\lim_{x\to{\pi ++}}\left(\frac{\sqrt{x}}{\csc\left(x\right)}\right)$
2

Adding $$ and $$

$\lim_{x\to{\pi +2}}\left(\frac{\sqrt{x}}{\csc\left(x\right)}\right)$
3

As the limit results in indeterminate form, we can apply L'Hôpital's rule

$\lim_{x\to{\pi +2}}\left(\frac{\frac{d}{dx}\left(\sqrt{x}\right)}{\frac{d}{dx}\left(\csc\left(x\right)\right)}\right)$
4

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\lim_{x\to{\pi +2}}\left(\frac{\frac{1}{2}x^{-\frac{1}{2}}}{\frac{d}{dx}\left(\csc\left(x\right)\right)}\right)$
5

Taking the derivative of cosecant

$\lim_{x\to{\pi +2}}\left(\frac{\frac{1}{2}x^{-\frac{1}{2}}}{-\cot\left(x\right)\csc\left(x\right)}\right)$
6

As the limit results in indeterminate form, we can apply L'Hôpital's rule

$\lim_{x\to{\pi +2}}\left(\frac{\frac{d}{dx}\left(\frac{1}{2}x^{-\frac{1}{2}}\right)}{\frac{d}{dx}\left(-\cot\left(x\right)\csc\left(x\right)\right)}\right)$
7

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=-\csc\left(x\right)$ and $g=\cot\left(x\right)$

$\lim_{x\to{\pi +2}}\left(\frac{\frac{d}{dx}\left(\frac{1}{2}x^{-\frac{1}{2}}\right)}{\cot\left(x\right)\frac{d}{dx}\left(-\csc\left(x\right)\right)-\frac{d}{dx}\left(\cot\left(x\right)\right)\csc\left(x\right)}\right)$
8

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=-1$ and $g=\csc\left(x\right)$

$\lim_{x\to{\pi +2}}\left(\frac{\frac{d}{dx}\left(\frac{1}{2}x^{-\frac{1}{2}}\right)}{\cot\left(x\right)\left(\csc\left(x\right)\frac{d}{dx}\left(-1\right)-\frac{d}{dx}\left(\csc\left(x\right)\right)\right)-\frac{d}{dx}\left(\cot\left(x\right)\right)\csc\left(x\right)}\right)$
9

The derivative of the constant function is equal to zero

$\lim_{x\to{\pi +2}}\left(\frac{\frac{d}{dx}\left(\frac{1}{2}x^{-\frac{1}{2}}\right)}{\cot\left(x\right)\left(0\csc\left(x\right)-\frac{d}{dx}\left(\csc\left(x\right)\right)\right)-\frac{d}{dx}\left(\cot\left(x\right)\right)\csc\left(x\right)}\right)$
10

Any expression multiplied by $0$ is equal to $0$

$\lim_{x\to{\pi +2}}\left(\frac{\frac{d}{dx}\left(\frac{1}{2}x^{-\frac{1}{2}}\right)}{\cot\left(x\right)\left(0-\frac{d}{dx}\left(\csc\left(x\right)\right)\right)-\frac{d}{dx}\left(\cot\left(x\right)\right)\csc\left(x\right)}\right)$
11

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\frac{1}{2}$ and $g=x^{-\frac{1}{2}}$

$\lim_{x\to{\pi +2}}\left(\frac{\frac{1}{2}\cdot\frac{d}{dx}\left(x^{-\frac{1}{2}}\right)+x^{-\frac{1}{2}}\cdot\frac{d}{dx}\left(\frac{1}{2}\right)}{\cot\left(x\right)\left(0-\frac{d}{dx}\left(\csc\left(x\right)\right)\right)-\frac{d}{dx}\left(\cot\left(x\right)\right)\csc\left(x\right)}\right)$
12

The derivative of the constant function is equal to zero

$\lim_{x\to{\pi +2}}\left(\frac{\frac{1}{2}\cdot\frac{d}{dx}\left(x^{-\frac{1}{2}}\right)+0x^{-\frac{1}{2}}}{-\cot\left(x\right)\frac{d}{dx}\left(\csc\left(x\right)\right)-\frac{d}{dx}\left(\cot\left(x\right)\right)\csc\left(x\right)}\right)$
13

Any expression multiplied by $0$ is equal to $0$

$\lim_{x\to{\pi +2}}\left(\frac{\frac{1}{2}\cdot\frac{d}{dx}\left(x^{-\frac{1}{2}}\right)+0}{-\cot\left(x\right)\frac{d}{dx}\left(\csc\left(x\right)\right)-\frac{d}{dx}\left(\cot\left(x\right)\right)\csc\left(x\right)}\right)$
14

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$\lim_{x\to{\pi +2}}\left(\frac{0-\frac{1}{4}x^{-\frac{3}{2}}}{-\cot\left(x\right)\frac{d}{dx}\left(\csc\left(x\right)\right)-\frac{d}{dx}\left(\cot\left(x\right)\right)\csc\left(x\right)}\right)$
15

Taking the derivative of cotangent

$\lim_{x\to{\pi +2}}\left(\frac{0-\frac{1}{4}x^{-\frac{3}{2}}}{\csc\left(x\right)^2\csc\left(x\right)-\cot\left(x\right)\frac{d}{dx}\left(\csc\left(x\right)\right)}\right)$
16

Taking the derivative of cosecant

$\lim_{x\to{\pi +2}}\left(\frac{0-\frac{1}{4}x^{-\frac{3}{2}}}{\csc\left(x\right)^2\csc\left(x\right)+\cot\left(x\right)\cot\left(x\right)\csc\left(x\right)}\right)$
17

$x+0=x$, where $x$ is any expression

$\lim_{x\to{\pi +2}}\left(\frac{-\frac{1}{4}x^{-\frac{3}{2}}}{\csc\left(x\right)^2\csc\left(x\right)+\cot\left(x\right)\cot\left(x\right)\csc\left(x\right)}\right)$
18

When multiplying exponents with same base you can add the exponents

$\lim_{x\to{\pi +2}}\left(\frac{-\frac{1}{4}x^{-\frac{3}{2}}}{\csc\left(x\right)^{3}+\cot\left(x\right)\cot\left(x\right)\csc\left(x\right)}\right)$
19

When multiplying exponents with same base you can add the exponents

$\lim_{x\to{\pi +2}}\left(\frac{-\frac{1}{4}x^{-\frac{3}{2}}}{\csc\left(x\right)^{3}+\cot\left(x\right)^2\csc\left(x\right)}\right)$
20

Factoring by $\csc\left(x\right)$

$\lim_{x\to{\pi +2}}\left(\frac{-\frac{1}{4}x^{-\frac{3}{2}}}{\left(\cot\left(x\right)^2+\csc\left(x\right)^{2}\right)\csc\left(x\right)}\right)$
21

Evaluating the limit when $x$ tends to ${\pi +2}$

$\frac{\left(\pi +2\right)^{-1.5}\left(-0.25\right)}{\left(\cot\left(\pi +2\right)^2+\csc\left(\pi +2\right)^{2}\right)\cdot \csc\left(\pi +2\right)}$
22

Simplifying

$\frac{\left(\pi +2\right)^{-1.5}\left(-0.25\right)}{\left(\cot\left(\pi +2\right)^2+\csc\left(\pi +2\right)^{2}\right)\cdot \csc\left(\pi +2\right)}$
23

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$\frac{\frac{1}{\sqrt{\left(\pi +2\right)^{3}}}\left(-0.25\right)}{\left(\cot\left(\pi +2\right)^2+\csc\left(\pi +2\right)^{2}\right)\cdot \csc\left(\pi +2\right)}$
24

Apply the formula: $a\frac{1}{x}$$=\frac{a}{x}$, where $a=-\frac{1}{4}$ and $x=\sqrt{\left(\pi +2\right)^{3}}$

$\frac{\frac{-0.25}{\sqrt{\left(\pi +2\right)^{3}}}}{\left(\cot\left(\pi +2\right)^2+\csc\left(\pi +2\right)^{2}\right)\cdot \csc\left(\pi +2\right)}$
25

Applying the trigonometric identity: $\cot\left(\theta\right)=\frac{1}{\tan\left(\theta\right)}$

$\frac{\frac{-0.25}{\sqrt{\left(\pi +2\right)^{3}}}}{\left(\left(\frac{1}{\tan\left(\pi +2\right)}\right)^2+\csc\left(\pi +2\right)^{2}\right)\cdot \csc\left(\pi +2\right)}$
26

The power of a quotient is equal to the quotient of the power of the numerator and denominator: $\displaystyle\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}$

$\frac{\frac{-0.25}{\sqrt{\left(\pi +2\right)^{3}}}}{\left(\frac{1}{\tan\left(\pi +2\right)^2}+\csc\left(\pi +2\right)^{2}\right)\cdot \csc\left(\pi +2\right)}$
27

Applying the trigonometric identity: $\cot\left(\theta\right)=\frac{1}{\tan\left(\theta\right)}$

$\frac{\frac{-0.25}{\sqrt{\left(\pi +2\right)^{3}}}}{\left(\cot\left(\pi +2\right)^2+\csc\left(\pi +2\right)^{2}\right)\cdot \csc\left(\pi +2\right)}$

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