Tell me, do you?

The limit given can also be written as follows: .
Given this, let .
The limit can now again be rewritten as follows: .
Using the difference of powers formula, we write this limit out again and complete the solution:

Let $f(x):=x^{123456789}$ . By definition of derivative at $x=1$ , $f'(1)=\lim_{x\to 1}\frac{(1+h)^{123456789}-1}{h}.$ Therefore, $\lim_{x\to 1}\frac{(1+h)^{123456789}-1}{h}=123456789.$

When the value of h is put in the expression ,we get the expression in the form 0/0. So if we apply L hopitals rule , we get the expression in the form 123456789(1+h)^123456788 Applying limit we get answer as 123456789

This is a great problem, but seems like all the methods involve taking a derivative, but how would you do it if you have not derived the formula in the first place?

We note that $(1+h)^n = 1+nh + \frac{n(n-1)}{2}h^2 +...+h^n$

So subtracting 1 and dividing by $h$ gives

$n +$ terms with $h, h^2 ... h^{n-1}$

Thus, taking $h$ to zero gives $n$ for sufficiently small $h$ .

This represents the derivative of the function $f(x)={ x }^{ 123456789 }$ at x = 1 so $f^{ ' }\left( 1 \right) =123456789{ (1) }^{ 123456789-1 }=123456789$

Apply La Hospital rule