Limits doubt(conceptual)

[wanton vessel]

Can anyone help pls?

[sour ivy]

1. Wait patiently for a helper to come along.
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[open quarry]

im sorry, what is this? HAHAH
there are one sided limits, one where x -> a from the left side (negative side) or the right side (pos. side), for instance you can take the lim x -> 0 1/x from the left side, which we'll denote it as the lim x -> 0- 1/x (the - is a superscript). so you can try substitution but for x values approaching 0 from the negative side, so 1/-0.1, or 1/-0.001, or 1/-0.00001, etc

or you can go from the right side (approaching the value from the pos. side) so you can approach zero from the positive numbers, 1/0.1, 1/0.001, and so forth

if the lim x -> a+ f(x) = lim x -> a- f(x) then the lim x -> a f(x) does not exist

[wanton vessel]

thats the opposite of what i learnt
i thought if they were equal they exist?

[open quarry]

oh for example if you took the lim x->0+ 1/x you get positive infinity

while if you took the lim x->0- 1/x you get negative infinity

so the lim x->0 1/x = DNE

cause if you look at the graph of 1/x you can see how it approaxhes pos. infinity and neg. infinity at the same time

[wanton vessel]

"if the lim x -> a+ f(x) = lim x -> a- f(x) then the lim x -> a f(x) does not exist" is what u told
That says if they were equal they do not exist?

[open quarry]

oh i wrote /=

which is not equal to

so if the right sided limit does not equal to the left sided limit then the limit does not exist!

[wanton vessel]

ok,but can u answer my original question
i think the answer is right side but i need confirmation

[open quarry]

ah yeah thats where i was confused on which one are we supposed to answer? HAHAH

it says here h is approaching 0 from the right / left side of 0 but which..??

[wanton vessel]

yeah,thats my original question

(that first line in the image is from my textbook)

[open quarry]

is it asking about these??

[wanton vessel]

yea,over there it is given h->0
here we have to take h approaching 0 from positive or negative side,it is not specified in my book

[wary vale]

both h should be strictly positive

if you assume $h>0$ for both, then $\lim_{h\to 0}f(a+h)=\lim_{x\to a^+}f(x)$ and $\lim_{h\to 0}f(a-h)=\lim_{x\to a^-}f(x)$

[strong minnow]

omegabet_

[wanton vessel]

thanks,got it

+close