Proving the Limit of a Complicated Function using Variables and Compositions

[Hygelac]

I'm stuck on one complicated limits problem, wondering if any of you could help me :) usually I am pretty fine with limits but this one uses all variables and has functions in it. Anyways, here it is:

f(a) = b, g(b) = c, h(c) = d
prove lim[x->a](h°g°f)(a) = d

(° = "of")

Can anyone help me?

 

[vsage]

lim[x->a](h(g(f(x)))) = d eh?

Well as x->a, f(x)->b
let y = f(x)
So now we have lim[y->b](h(g(y)))

as y->b, g(y)->c

Make sense? I guess the rest is obvious

 

[Hygelac]

Thanks for your help, I just thought of something else, too. Seems like this works and is very easy, could you do just a quick check of it and see if it makes sense?

f(a) = b, g(b) = c, h(c) = d

lim[x->a](h(g(f(a)))) = d

f(a) = b, so g(f(a)) = g(b)
g(b) = c, so h(g(b)) = h(c)
h(c) = d, which solves the problem

 

[vsage]

Looks good to me. You may want to play it safe I'm not sure how meticulous your teacher expects you to me but I would accept that answer :)

 

[Hygelac]

Thanks a bunch :)