# Use definition of limit to prove $\lim_ {t \to a} f(t^2) =L$

Suppose:

$$\lim_ {x \to a^2} f(x) =L$$

Use this to prove $$\lim_{t \to a} f(t^2) =L$$.

This is evident from the limit substitution rule, but in this course we have not covered this and we should use the formal definition of the limit. That is:

$$\forall \epsilon>0 \quad \exists \delta \quad s.t. \quad |x-a^2|< \delta \implies |f(x)-L|< \epsilon$$

must be used to prove:

$$\forall \epsilon>0 \quad \exists \delta \quad s.t. \quad |t-a|< \delta \implies |f(t^2)-L|< \epsilon$$

how do I make this connection.

• Maybe this can follow from a more general result which is not so hard to prove… (substitution theorem for limits) – user408856 Jan 28 at 9:46
• What is that $\displaystyle\lim_{n\to\infty}$ doing in the title? – José Carlos Santos Jan 28 at 9:46

$$|t-a| <\delta'$$ implies $$|t^{2}-a^{2}|=|t-a||t+a| <\delta' (|t|+|a|)<\delta' (\delta'+2|a|)$$. So choose $$\delta'$$ such that $$\delta' (\delta'+2|a|) <\delta$$. It is enough to take $$\delta' <1$$ and $$\delta' <\frac {\delta} {1+2|a|}$$. Then $$|t-a| <\delta'$$ implies $$|f(t^{2})-L| <\epsilon$$.
• note to self: set $x=t^2$ and it follows. – Wesley Strik Jan 28 at 10:04