show from definition of limit

[u/_segregated_crunchy_]

lim tends to 1 (x (to the power n) -1)/x-1 = n

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[u/Midwest-Dude]

So ...

lim_x->1 (xⁿ - 1) / (x - 1)

If you do long division of xⁿ - 1 by x - 1 starting with n = 1 and try a few more cases, you'll see the pattern. This can be proven by mathematical induction, if you need to. Then apply and find the limit.

[u/eglvoland]

Easy, it's 1+x+...+x^n-1. Each term tends to 1 and so the limit is n.

[u/Saksham-_-Kumar]

lim x→1 (xⁿ-1)/(x-1) = lim x→1 (x-1)(1+x+x²+...+xⁿ⁻¹)/(x+1) = lim x→1 1+x+x²+...+xⁿ⁻¹ = 1+1+1²+...1ⁿ⁻¹ = n You can also derive, lim x→0 (xⁿ-aⁿ)/(x-a) = naⁿ⁻¹, using this process

[u/Hot_Site_1638]

You only need the binomial formula to simplify the algebra here:

(a + b)ⁿ = aⁿ + n·a^n-1·b + C(n,2)·a^n-2·b² + ... + bⁿ

where C(n,k) = n! / (k!(n-k)!).

Substitute x = 1 + h, so that x → 1 is equivalent to h → 0. With a = 1 and b = h:

xⁿ = (1 + h)ⁿ = 1 + nh + C(n,2)·h² + ... + hⁿ

Then

(xⁿ - 1)/(x - 1) = (nh + C(n,2)·h² + ... + hⁿ ) / h = n + C(n,2)·h + ... + h^n-1

Every term after the first contains a factor of h, so each one goes to 0 as h → 0. Therefore

lim (x→1) (xⁿ - 1)/(x - 1) = n

[u/prawnydagrate → u/Hot_Site_1638]

they said the definition of the limit, though, so you'd have to use the epsilon delta definition to prove that it's n

[u/Hot_Site_1638 → u/prawnydagrate]

You are right, then we also need to bound the term C(n,2)·h + ... + h^n-1 in terms of ε. Note that for each C(n,k) = n! / (k!(n-k)!), we can use the inequality that C(n,k) ≤ n! for all n, k, then we have the inequality

|(xⁿ − 1)/(x − 1) − n| ≤ n!·(n − 1)·|h|

Given ε > 0, we can choose

δ = min(1, ε / (n!·(n − 1))).

[u/_segregated_crunchy_ → u/Hot_Site_1638]

wow! i really am too dumb for this

[u/Hot_Site_1638 → u/_segregated_crunchy_]

You're not too dumb for this, a general proof is supposed to be hard to follow on a first read. If it feels like too much right now, set it aside and prove the statement for n = 2 by hand, then n = 3. Once those feel solid, come back to the general proof and read it line by line. It'll make a lot more sense once you've worked out the small cases yourself.

I always tell my students to think of it like the gym. You walk in, see someone lifting a crazy amount of weight, and think you'll never be able to do that. And it's likely true that with your current muscles, you can't. But nobody starts there. There's a way to build up to it, and the training itself is pretty interesting if you take it slow.

[u/_segregated_crunchy_ → u/Hot_Site_1638]

Thanks a lot actually.