Difficult limit problem calculus

[terse dove]

Need to solve it using l'hospital btw

[runic pasture]

take the natural log, and see what you get

$e^{\lim_{t\to\infty}t\ln\left(1-\frac{3}{t}\right)}$

[hybrid ice]

k

[runic pasture]

now, find the limit of the exponent

[terse dove]

Can you just take the log of a limit and it remains the same?

Confused how you did that

Oh wait, it is e raised to the limit

Thats really weird, let me see if I can figure that out though

[runic pasture]

e^(ln(x))=x

[terse dove]

Thanks for the help @runic pasture !

[timber kestrel]

@terse dove has given 1 rep to @runic pasture

[boreal vortex]

This uses the composition law for limits

if lim_x->c g(x)=G, then lim_x->c f(g(x))= lim_x->G f(x) = L

So if f is continuous at L, L = f(G). In this specific example f would be e^t and g would be tln(1+1/t)

This is a very powerful result with limits and is worth spending time on understanding

@terse dove

As for how to evaluate the new limit, tln(1+1/t) is the same as ln(1+1/t)/(1/t)

Evaluating this at infinity gives you 0/0 so lhoptials rule is applicable