Good Evening everyone, I just wrote my final examination for analysis. One of the questions was one that we had in a previous quiz, the solution of which I still don't quite understand. The question is as see below. Could someone please elucidate answer. I don't know how, but there was some confusion in the forum by the wording so, I state clearly: As you can see in the picture. the answer is C). I understand why III) is wrong and II) is correct. but by the definition of a limit: $$ for\, every \, \varepsilon > 0 \, there\, exists \, \delta >0 \, such \, that$$ $$0<|x-a| < \delta\, \rightarrow |f(x) - L| <\varepsilon $$ we then have that $$-\varepsilon < f(x) - L < \varepsilon \rightarrow L-\varepsilon <f(x)<L+\varepsilon$$ and thus f(x) is in $(L-\varepsilon , L+\varepsilon)$
so I don't get why I) is wrong.
Thank you.
What is wrong with I) is that the interval for $x$ includes the case $x=a$ whereas the limit definition excludes $x=a$. Why is this important? Consider the case $f(x)=sin(x)/x$ and $L=0$.