# Limits Calculator

## Get detailed solutions to your math problems with our Limits step by step calculator. Sharpen your math skills and learn step by step with our math solver. Check out more online calculators here.

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### Difficult Problems

1

Solved example of Limits by L'Hôpital's rule

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

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

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

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

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

The derivative of the linear function times a constant, is equal to the constant

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

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 ${f'(x) = sec^2(x)\cdot D_x(x)}$

$\lim_{x\to0}\left(\frac{3\cos\left(3x\right)}{\sec\left(4x\right)^2\frac{d}{dx}\left(4x\right)}\right)$
6

The derivative of the linear function times a constant, is equal to the constant

$\lim_{x\to0}\left(\frac{3\cos\left(3x\right)}{4\sec\left(4x\right)^2}\right)$
7

Applying the trigonometric identity: $\displaystyle\frac{1}{\sec^{n}(\theta)}=\cos^{n}(\theta)$

$\lim_{x\to0}\left(\frac{3\cos\left(4x\right)^2\cos\left(3x\right)}{4}\right)$
8

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

$\frac{1}{4}\lim_{x\to0}\left(3\cos\left(4x\right)^2\cos\left(3x\right)\right)$
9

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

$\frac{1}{4}\cos\left(3\cdot 0\right)3\cos\left(4\cdot 0\right)^2$
10

Simplifying

$\frac{3}{4}$