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Difficult Problems

1

Example

$\lim_{x\to0}\left(\frac{\sin\left(3x\right)}{4x}\right)$
2

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

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(\sin\left(3x\right)\right)}{\frac{d}{dx}\left(4x\right)}\right)$
3

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(\sin\left(3x\right)\right)}{4\frac{d}{dx}\left(x\right)}\right)$
4

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

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(\sin\left(3x\right)\right)}{4}\right)$
5

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 ${f'(x) = \cos(x)\cdot D_x(x)}$

$\lim_{x\to0}\left(\frac{\frac{d}{dx}\left(3x\right)\cos\left(3x\right)}{4}\right)$
6

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$\lim_{x\to0}\left(\frac{3\cos\left(3x\right)\frac{d}{dx}\left(x\right)}{4}\right)$
7

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

$\lim_{x\to0}\left(\frac{3\cos\left(3x\right)}{4}\right)$
8

The limit of the product of a function and a constant is equal to the limit of the function, times the constant: $\displaystyle \lim_{t\to 0}{\left(2t\right)}=2\cdot\lim_{t\to 0}{\left(t\right)}$

$\frac{1}{4}\lim_{x\to0}\left(3\cos\left(3x\right)\right)$
9

Evaluating the limit when $x$ tends to $0$

$\cos\left(0\cdot 3\right)\cdot 3\cdot \frac{1}{4}$
10

Simplifying

$\frac{3}{4}$

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