Limits

Question

How to slove this

$\lim_{n\rightarrow \infty }\left ( 10^{n}+n^{20} \right )/n!$

Comments

Its $0$. See this. But still figuring out a mathematical way to do this.

Answer 1

$\lim_{n\rightarrow \infty }\left ( 10^{n}+n^{20} \right )/n!$

Say n=1/z

$\lim_{z\rightarrow 0 }\left ( 10^{1/z}+(\frac{1}{z})^{20} \right )/\frac{1}{z!}$

=$\lim_{z\rightarrow 0 }z!\left ( 10^{1/z}+1 \right )/z^{20}$

=1/0 [putting z=0]=$\alpha$

then for$\underset{n \to \alpha}{lim}$ value will be 0

Comments

In LINE4 u cant takout z power 20 without taking lcm ...

Answer 2

Another Take on how this problem can be interpreted and solved

Break this problem into Numerator and Denominator

That is numerator (10^n + n^20 ) and Denominator is n!

(10^n + n^20 ) is exponential in nature (sum of 2 exponential operands).

The thing to note here is denominator "grows faster" than that of the numerator in this case(factorials grow faster than exponents like example Constant^n or n^constant) .

Hence it starts suppressing the value and as the limit tends to infinity f(x) will tend towards Zero.