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To prove
$\lim_{x\to0^+} f(x)= \lim_{x\to0^-} f(-x)$

I am trying to make my fundamentals strong in limits which is why I am solving the Spivak's book. Now, this problem is seemingly very simple but I'm stuck in the following way:

Let $\lim_{x\to0^+} f(x)= L$. Then according to the definition given in spivak calculus for "Limits from above", given $\epsilon \gt0$ there is $\delta \gt 0$, such that for all $x$ if $0 \lt x \lt \delta$ then $|f(x)-L|\lt\epsilon$.

Now according to the definition given in spivak calculus for "Limits from below", if, $\lim_{x\to0^-} f(-x)= L$, then given $\epsilon \gt0$ there is $\delta' \gt 0$, such that for all $x$ if $0 \lt -x \lt \delta'$ then $|f(-x)-L|\lt\epsilon$

I tried putting $y=-x$ in the definition of "limits from above" but couldn't arrive at the definition for "limits from below".

Any help in solving this problem to clear my concept on limits is appreciated, Thank you.

José Carlos Santos
  • 427,504
aditya bhatt
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1 Answers1

1

Suppose that $\lim_{x\to0^+}f(x)=L$. Take $\varepsilon>0$. Then, as you know, there is a $\delta>0$ such that$$0<x<\delta\implies\bigl\lvert f(x)-L\bigr\rvert<\varepsilon.$$But then$$-\delta<x<0\implies\bigl\lvert f(-x)-L\bigr\rvert<\varepsilon.$$So, I proved that$$\lim_{x\to0^+}f(x)=L\implies\lim_{x\to0^-}f(x)=L$$and, by the same argument,$$\lim_{x\to0^-}f(x)=L\implies\lim_{x\to0^+}f(x)=L.$$

José Carlos Santos
  • 427,504
  • Here, at $-\delta<x<0\implies\bigl\lvert f(-x)-L\bigr\rvert<\varepsilon$, do you mean $-\delta<-x<0\implies\bigl\lvert f(-x)-L\bigr\rvert<\varepsilon$? If not, are we allowed to change the independent variable of the function this way? – aditya bhatt Oct 19 '19 at 07:26
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    What I wrote is what I meant. If you want more details:\begin{align}-\delta<x<0&\implies0<-x<\delta\&\implies\bigl\lvert f(-x)-L\bigr\rvert<\varepsilon.\end{align} – José Carlos Santos Oct 19 '19 at 07:35
  • Sir, thank you for giving me this insight for proof. I used this insight in another problem(problem 34) and could prove the required question and I'm sure it will be useful in many other problems. Thanks a lot! – aditya bhatt Oct 19 '19 at 14:42
  • I'm glad I could help. – José Carlos Santos Oct 19 '19 at 14:43
  • Sir, as I am working to make my fundamentals of calculus very strong, what other books would you suggest me to solve the problems of so that my fundamentals of calculus and proving techniques become strong? Also, thank you for taking your time to answer my question. – aditya bhatt Oct 19 '19 at 15:13
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    I suggest no other book. Spivak's Calculus is the best one, as far as I am concerned. – José Carlos Santos Oct 19 '19 at 15:16