# Limits by direct substitution Calculator

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### Difficult Problems

1

Example

$\lim_{x\to∞}\left(\frac{x^{2\log\left(x\right)}}{\left(x\cdot \log\left(x\right)\right)^3}\right)$
2

Calculating the natural logarithm of $2$

$\lim_{x\to∞}\left(\frac{x^{\frac{4}{\sqrt{3}}\cdot 10x\cdot n}}{\left(10x\cdot n\ln\left(x\right)\right)^3}\right)$
3

Multiply $10$ times $\frac{4}{\sqrt{3}}$

$\lim_{x\to∞}\left(\frac{x^{\sqrt{48}x\cdot n}}{\left(10x\cdot n\ln\left(x\right)\right)^3}\right)$
4

The power of a product is equal to the product of it's factors raised to the same power

$\lim_{x\to∞}\left(\frac{x^{\sqrt{48}x\cdot n}}{1000x^3\ln\left(x\right)^3n^3}\right)$
5

Taking out the constant $1000$ from the fraction's denominator

$\lim_{x\to∞}\left(\frac{1}{1000}\cdot\frac{x^{\sqrt{48}x\cdot n}}{x^3\ln\left(x\right)^3n^3}\right)$
6

The limit of the product of two functions is equal to the product of the limits of each function

$\lim_{x\to∞}\left(\frac{1}{1000}\right)\lim_{x\to∞}\left(\frac{x^{\sqrt{48}x\cdot n}}{x^3\ln\left(x\right)^3n^3}\right)$
7

The limit of a constant is just the constant

$\frac{1}{1000}\lim_{x\to∞}\left(\frac{x^{\sqrt{48}x\cdot n}}{x^3\ln\left(x\right)^3n^3}\right)$
8

The limit of quotient of two functions is the quotient of their limits

$\frac{1}{1000}\cdot\frac{\lim_{x\to∞}\left(x^{\sqrt{48}x\cdot n}\right)}{\lim_{x\to∞}\left(x^3\ln\left(x\right)^3n^3\right)}$
9

Multiplying the fraction and term

$\frac{\frac{1}{1000}\lim_{x\to∞}\left(x^{\sqrt{48}x\cdot n}\right)}{\lim_{x\to∞}\left(x^3\ln\left(x\right)^3n^3\right)}$
10

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

$\frac{\frac{1}{1000}\lim_{x\to∞}\left(x^{\sqrt{48}x\cdot n}\right)}{n^3\lim_{x\to∞}\left(x^3\ln\left(x\right)^3\right)}$
11

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

$\frac{\frac{1}{1000}\lim_{x\to∞}\left(x^{\sqrt{48}x\cdot n}\right)}{n^3∞^3\ln\left(∞\right)^3}$
12

Simplifying

$\frac{\frac{1}{1000}\lim_{x\to∞}\left(x^{\sqrt{48}x\cdot n}\right)}{n^3∞^3\ln\left(∞\right)^3}$
13

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

$\frac{\frac{1}{1000}∞^{\sqrt{48}∞n}}{n^3∞^3\ln\left(∞\right)^3}$
14

Simplifying

$\frac{\frac{1}{1000}∞^{\sqrt{48}∞n}}{n^3∞^3\ln\left(∞\right)^3}$
15

Rewriting the exponent

$\frac{\frac{1}{1000}\left(∞^{\sqrt{48}n}\right)^∞}{n^3∞^3\ln\left(∞\right)^3}$
16

Applying the power of a power property

$\frac{\frac{1}{1000}∞^{\sqrt{48}∞n}}{n^3∞^3\ln\left(∞\right)^3}$