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1

Solved example of limits by l'hôpital's rule

$\lim_{x\to\pi}\left(\frac{\tan\left(1+\cos\left(x\right)\right)}{\cos\left(\tan\left(x\right)\right)-1}\right)$
2

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

$\lim_{x\to\pi }\left(\frac{\frac{d}{dx}\left(\tan\left(1+\cos\left(x\right)\right)\right)}{\frac{d}{dx}\left(\cos\left(\tan\left(x\right)\right)-1\right)}\right)$
3

The derivative of the tangent of a function is equal to secant squared of that function times the derivative of that function, in other words, if ${f(x) = tan(x)}$, then ${f'(x) = sec^2(x)\cdot D_x(x)}$

$\lim_{x\to\pi }\left(\frac{\sec\left(1+\cos\left(x\right)\right)^2\frac{d}{dx}\left(1+\cos\left(x\right)\right)}{\frac{d}{dx}\left(\cos\left(\tan\left(x\right)\right)-1\right)}\right)$
4

The derivative of a sum of two functions is the sum of the derivatives of each function

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

The derivative of the constant function ($-1$) is equal to zero

$\lim_{x\to\pi }\left(\frac{\sec\left(1+\cos\left(x\right)\right)^2\left(0+\frac{d}{dx}\left(\cos\left(x\right)\right)\right)}{\frac{d}{dx}\left(\cos\left(\tan\left(x\right)\right)\right)}\right)$
6

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

$\lim_{x\to\pi }\left(\frac{\sec\left(1+\cos\left(x\right)\right)^2\frac{d}{dx}\left(\cos\left(x\right)\right)}{\frac{d}{dx}\left(\cos\left(\tan\left(x\right)\right)\right)}\right)$
7

The derivative of the cosine of a function is equal to minus the sine of the function times the derivative of the function, in other words, if $f(x) = \cos(x)$, then $f'(x) = -\sin(x)\cdot D_x(x)$

$\lim_{x\to\pi }\left(\frac{-\sec\left(1+\cos\left(x\right)\right)^2\sin\left(x\right)}{\frac{d}{dx}\left(\cos\left(\tan\left(x\right)\right)\right)}\right)$
8

The derivative of the cosine of a function is equal to minus the sine of the function times the derivative of the function, in other words, if $f(x) = \cos(x)$, then $f'(x) = -\sin(x)\cdot D_x(x)$

$\lim_{x\to\pi }\left(\frac{-\sec\left(1+\cos\left(x\right)\right)^2\sin\left(x\right)}{-\sin\left(\tan\left(x\right)\right)\frac{d}{dx}\left(\tan\left(x\right)\right)}\right)$
9

Apply the formula: $\frac{a}{bx}$$=\frac{\frac{a}{b}}{x}$, where $a=-1$, $b=-1$ and $x=\sin\left(\tan\left(x\right)\right)\frac{d}{dx}\left(\tan\left(x\right)\right)$

$\lim_{x\to\pi }\left(\frac{\sec\left(1+\cos\left(x\right)\right)^2\sin\left(x\right)}{\sin\left(\tan\left(x\right)\right)\frac{d}{dx}\left(\tan\left(x\right)\right)}\right)$
10

The derivative of the tangent of a function is equal to secant squared of that function times the derivative of that function, in other words, if ${f(x) = tan(x)}$, then ${f'(x) = sec^2(x)\cdot D_x(x)}$

$\lim_{x\to\pi }\left(\frac{\sec\left(1+\cos\left(x\right)\right)^2\sin\left(x\right)}{\sec\left(x\right)^2\sin\left(\tan\left(x\right)\right)\frac{d}{dx}\left(x\right)}\right)$
11

The derivative of the linear function is equal to $1$

$\lim_{x\to\pi }\left(\frac{\sec\left(1+\cos\left(x\right)\right)^2\sin\left(x\right)}{\sec\left(x\right)^2\sin\left(\tan\left(x\right)\right)}\right)$
12

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

$\lim_{x\to\pi }\left(\frac{\frac{d}{dx}\left(\sec\left(1+\cos\left(x\right)\right)^2\sin\left(x\right)\right)}{\frac{d}{dx}\left(\sec\left(x\right)^2\sin\left(\tan\left(x\right)\right)\right)}\right)$
13

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\sec\left(1+\cos\left(x\right)\right)^2$ and $g=\sin\left(x\right)$

$\lim_{x\to\pi }\left(\frac{\frac{d}{dx}\left(\sec\left(1+\cos\left(x\right)\right)^2\right)\sin\left(x\right)+\sec\left(1+\cos\left(x\right)\right)^2\frac{d}{dx}\left(\sin\left(x\right)\right)}{\frac{d}{dx}\left(\sec\left(x\right)^2\sin\left(\tan\left(x\right)\right)\right)}\right)$
14

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=\sin\left(\tan\left(x\right)\right)$ and $g=\sec\left(x\right)^2$

$\lim_{x\to\pi }\left(\frac{\frac{d}{dx}\left(\sec\left(1+\cos\left(x\right)\right)^2\right)\sin\left(x\right)+\sec\left(1+\cos\left(x\right)\right)^2\frac{d}{dx}\left(\sin\left(x\right)\right)}{\sec\left(x\right)^2\frac{d}{dx}\left(\sin\left(\tan\left(x\right)\right)\right)+\sin\left(\tan\left(x\right)\right)\frac{d}{dx}\left(\sec\left(x\right)^2\right)}\right)$
15

The derivative of the sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if ${f(x) = \sin(x)}$, then ${f'(x) = \cos(x)\cdot D_x(x)}$

$\lim_{x\to\pi }\left(\frac{\frac{d}{dx}\left(\sec\left(1+\cos\left(x\right)\right)^2\right)\sin\left(x\right)+\sec\left(1+\cos\left(x\right)\right)^2\cos\left(x\right)}{\sec\left(x\right)^2\frac{d}{dx}\left(\sin\left(\tan\left(x\right)\right)\right)+\sin\left(\tan\left(x\right)\right)\frac{d}{dx}\left(\sec\left(x\right)^2\right)}\right)$
16

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

$\lim_{x\to\pi }\left(\frac{2\sec\left(1+\cos\left(x\right)\right)^{\left(2-1\right)}\cdot\frac{d}{dx}\left(\sec\left(1+\cos\left(x\right)\right)\right)\sin\left(x\right)+\sec\left(1+\cos\left(x\right)\right)^2\cos\left(x\right)}{\sec\left(x\right)^2\frac{d}{dx}\left(\sin\left(\tan\left(x\right)\right)\right)+2\sin\left(\tan\left(x\right)\right)\sec\left(x\right)\frac{d}{dx}\left(\sec\left(x\right)\right)}\right)$
17

Subtract the values $2$ and $-1$

$\lim_{x\to\pi }\left(\frac{2\sec\left(1+\cos\left(x\right)\right)^{1}\cdot\frac{d}{dx}\left(\sec\left(1+\cos\left(x\right)\right)\right)\sin\left(x\right)+\sec\left(1+\cos\left(x\right)\right)^2\cos\left(x\right)}{\sec\left(x\right)^2\frac{d}{dx}\left(\sin\left(\tan\left(x\right)\right)\right)+2\sin\left(\tan\left(x\right)\right)\sec\left(x\right)\frac{d}{dx}\left(\sec\left(x\right)\right)}\right)$
18

Any expression to the power of $1$ is equal to that same expression

$\lim_{x\to\pi }\left(\frac{2\sec\left(1+\cos\left(x\right)\right)\frac{d}{dx}\left(\sec\left(1+\cos\left(x\right)\right)\right)\sin\left(x\right)+\sec\left(1+\cos\left(x\right)\right)^2\cos\left(x\right)}{\sec\left(x\right)^2\frac{d}{dx}\left(\sin\left(\tan\left(x\right)\right)\right)+2\sin\left(\tan\left(x\right)\right)\sec\left(x\right)\frac{d}{dx}\left(\sec\left(x\right)\right)}\right)$
19

The derivative of the sine of a function is equal to the cosine of that function times the derivative of that function, in other words, if ${f(x) = \sin(x)}$, then ${f'(x) = \cos(x)\cdot D_x(x)}$

$\lim_{x\to\pi }\left(\frac{2\sec\left(1+\cos\left(x\right)\right)\frac{d}{dx}\left(\sec\left(1+\cos\left(x\right)\right)\right)\sin\left(x\right)+\sec\left(1+\cos\left(x\right)\right)^2\cos\left(x\right)}{\sec\left(x\right)^2\cos\left(\tan\left(x\right)\right)\frac{d}{dx}\left(\tan\left(x\right)\right)+2\sin\left(\tan\left(x\right)\right)\sec\left(x\right)\frac{d}{dx}\left(\sec\left(x\right)\right)}\right)$
20

The derivative of the tangent of a function is equal to secant squared of that function times the derivative of that function, in other words, if ${f(x) = tan(x)}$, then ${f'(x) = sec^2(x)\cdot D_x(x)}$

$\lim_{x\to\pi }\left(\frac{2\sec\left(1+\cos\left(x\right)\right)\frac{d}{dx}\left(\sec\left(1+\cos\left(x\right)\right)\right)\sin\left(x\right)+\sec\left(1+\cos\left(x\right)\right)^2\cos\left(x\right)}{\sec\left(x\right)^2\sec\left(x\right)^2\cos\left(\tan\left(x\right)\right)\frac{d}{dx}\left(x\right)+2\sin\left(\tan\left(x\right)\right)\sec\left(x\right)\frac{d}{dx}\left(\sec\left(x\right)\right)}\right)$
21

When multiplying exponents with same base we can add the exponents

$\lim_{x\to\pi }\left(\frac{2\sec\left(1+\cos\left(x\right)\right)\frac{d}{dx}\left(\sec\left(1+\cos\left(x\right)\right)\right)\sin\left(x\right)+\sec\left(1+\cos\left(x\right)\right)^2\cos\left(x\right)}{\sec\left(x\right)^{4}\cos\left(\tan\left(x\right)\right)\frac{d}{dx}\left(x\right)+2\sin\left(\tan\left(x\right)\right)\sec\left(x\right)\frac{d}{dx}\left(\sec\left(x\right)\right)}\right)$
22

The derivative of the linear function is equal to $1$

$\lim_{x\to\pi }\left(\frac{2\sec\left(1+\cos\left(x\right)\right)\frac{d}{dx}\left(\sec\left(1+\cos\left(x\right)\right)\right)\sin\left(x\right)+\sec\left(1+\cos\left(x\right)\right)^2\cos\left(x\right)}{\sec\left(x\right)^{4}\cos\left(\tan\left(x\right)\right)+2\sin\left(\tan\left(x\right)\right)\sec\left(x\right)\frac{d}{dx}\left(\sec\left(x\right)\right)}\right)$
23

Taking the derivative of secant

$\lim_{x\to\pi }\left(\frac{2\sec\left(1+\cos\left(x\right)\right)\sec\left(1+\cos\left(x\right)\right)\tan\left(1+\cos\left(x\right)\right)\frac{d}{dx}\left(1+\cos\left(x\right)\right)\sin\left(x\right)+\sec\left(1+\cos\left(x\right)\right)^2\cos\left(x\right)}{\sec\left(x\right)^{4}\cos\left(\tan\left(x\right)\right)+2\sin\left(\tan\left(x\right)\right)\sec\left(x\right)\frac{d}{dx}\left(\sec\left(x\right)\right)}\right)$
24

When multiplying exponents with same base you can add the exponents

$\lim_{x\to\pi }\left(\frac{2\sec\left(1+\cos\left(x\right)\right)^2\tan\left(1+\cos\left(x\right)\right)\frac{d}{dx}\left(1+\cos\left(x\right)\right)\sin\left(x\right)+\sec\left(1+\cos\left(x\right)\right)^2\cos\left(x\right)}{\sec\left(x\right)^{4}\cos\left(\tan\left(x\right)\right)+2\sin\left(\tan\left(x\right)\right)\sec\left(x\right)\frac{d}{dx}\left(\sec\left(x\right)\right)}\right)$
25

Taking the derivative of secant

$\lim_{x\to\pi }\left(\frac{2\sec\left(1+\cos\left(x\right)\right)^2\tan\left(1+\cos\left(x\right)\right)\frac{d}{dx}\left(1+\cos\left(x\right)\right)\sin\left(x\right)+\sec\left(1+\cos\left(x\right)\right)^2\cos\left(x\right)}{\sec\left(x\right)^{4}\cos\left(\tan\left(x\right)\right)+2\sin\left(\tan\left(x\right)\right)\sec\left(x\right)\sec\left(x\right)\tan\left(x\right)}\right)$
26

When multiplying exponents with same base you can add the exponents

$\lim_{x\to\pi }\left(\frac{2\sec\left(1+\cos\left(x\right)\right)^2\tan\left(1+\cos\left(x\right)\right)\frac{d}{dx}\left(1+\cos\left(x\right)\right)\sin\left(x\right)+\sec\left(1+\cos\left(x\right)\right)^2\cos\left(x\right)}{\sec\left(x\right)^{4}\cos\left(\tan\left(x\right)\right)+2\sec\left(x\right)^2\sin\left(\tan\left(x\right)\right)\tan\left(x\right)}\right)$
27

The derivative of a sum of two functions is the sum of the derivatives of each function

$\lim_{x\to\pi }\left(\frac{2\sec\left(1+\cos\left(x\right)\right)^2\tan\left(1+\cos\left(x\right)\right)\left(\frac{d}{dx}\left(1\right)+\frac{d}{dx}\left(\cos\left(x\right)\right)\right)\sin\left(x\right)+\sec\left(1+\cos\left(x\right)\right)^2\cos\left(x\right)}{\sec\left(x\right)^{4}\cos\left(\tan\left(x\right)\right)+2\sec\left(x\right)^2\sin\left(\tan\left(x\right)\right)\tan\left(x\right)}\right)$
28

Solve the product $2\sec\left(1+\cos\left(x\right)\right)^2\tan\left(1+\cos\left(x\right)\right)\left(\frac{d}{dx}\left(1\right)+\frac{d}{dx}\left(\cos\left(x\right)\right)\right)\sin\left(x\right)$

$\lim_{x\to\pi }\left(\frac{\sec\left(1+\cos\left(x\right)\right)^2\left(2\frac{d}{dx}\left(1\right)+2\frac{d}{dx}\left(\cos\left(x\right)\right)\right)\tan\left(1+\cos\left(x\right)\right)\sin\left(x\right)+\sec\left(1+\cos\left(x\right)\right)^2\cos\left(x\right)}{\sec\left(x\right)^{4}\cos\left(\tan\left(x\right)\right)+2\sec\left(x\right)^2\sin\left(\tan\left(x\right)\right)\tan\left(x\right)}\right)$
29

The derivative of the constant function ($1$) is equal to zero

$\lim_{x\to\pi }\left(\frac{\sec\left(1+\cos\left(x\right)\right)^2\left(2\cdot 0+2\frac{d}{dx}\left(\cos\left(x\right)\right)\right)\tan\left(1+\cos\left(x\right)\right)\sin\left(x\right)+\sec\left(1+\cos\left(x\right)\right)^2\cos\left(x\right)}{\sec\left(x\right)^{4}\cos\left(\tan\left(x\right)\right)+2\sec\left(x\right)^2\sin\left(\tan\left(x\right)\right)\tan\left(x\right)}\right)$
30

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

$\lim_{x\to\pi }\left(\frac{\sec\left(1+\cos\left(x\right)\right)^2\left(0+2\frac{d}{dx}\left(\cos\left(x\right)\right)\right)\tan\left(1+\cos\left(x\right)\right)\sin\left(x\right)+\sec\left(1+\cos\left(x\right)\right)^2\cos\left(x\right)}{\sec\left(x\right)^{4}\cos\left(\tan\left(x\right)\right)+2\sec\left(x\right)^2\sin\left(\tan\left(x\right)\right)\tan\left(x\right)}\right)$
31

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

$\lim_{x\to\pi }\left(\frac{2\sec\left(1+\cos\left(x\right)\right)^2\frac{d}{dx}\left(\cos\left(x\right)\right)\tan\left(1+\cos\left(x\right)\right)\sin\left(x\right)+\sec\left(1+\cos\left(x\right)\right)^2\cos\left(x\right)}{\sec\left(x\right)^{4}\cos\left(\tan\left(x\right)\right)+2\sec\left(x\right)^2\sin\left(\tan\left(x\right)\right)\tan\left(x\right)}\right)$
32

The derivative of the cosine of a function is equal to minus the sine of the function times the derivative of the function, in other words, if $f(x) = \cos(x)$, then $f'(x) = -\sin(x)\cdot D_x(x)$

$\lim_{x\to\pi }\left(\frac{-2\sec\left(1+\cos\left(x\right)\right)^2\sin\left(x\right)\tan\left(1+\cos\left(x\right)\right)\sin\left(x\right)+\sec\left(1+\cos\left(x\right)\right)^2\cos\left(x\right)}{\sec\left(x\right)^{4}\cos\left(\tan\left(x\right)\right)+2\sec\left(x\right)^2\sin\left(\tan\left(x\right)\right)\tan\left(x\right)}\right)$
33

When multiplying exponents with same base you can add the exponents

$\lim_{x\to\pi }\left(\frac{-2\sin\left(x\right)^2\sec\left(1+\cos\left(x\right)\right)^2\tan\left(1+\cos\left(x\right)\right)+\sec\left(1+\cos\left(x\right)\right)^2\cos\left(x\right)}{\sec\left(x\right)^{4}\cos\left(\tan\left(x\right)\right)+2\sec\left(x\right)^2\sin\left(\tan\left(x\right)\right)\tan\left(x\right)}\right)$
34

Evaluating the limit when $x$ tends to $\pi $

$\frac{\sin\left(\pi \right)^2-2\sec\left(1+\cos\left(\pi \right)\right)^2\tan\left(1+\cos\left(\pi \right)\right)+\sec\left(1+\cos\left(\pi \right)\right)^2\cos\left(\pi \right)}{\sec\left(\pi \right)^{4}\cos\left(\tan\left(\pi \right)\right)+\sec\left(\pi \right)^2\sin\left(\tan\left(\pi \right)\right)2\tan\left(\pi \right)}$
35

Simplifying

$-1$

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