Question

Can someone help me with evaluating these limits? *Attached below* Medal will be awarded :)

Answer 1

(a) \[\lim_{x \rightarrow 0-} \frac{ 1 }{ x }\]
(b) \[\lim_{x \rightarrow 0+}\frac{ 1 }{ x }\]
(c) \[\lim_{x \rightarrow 0}\frac{ 1 }{ x }\]
(d) \[\lim_{x \rightarrow 0}\frac{ |x| }{ x }\]

Answer 2

isn't 1/0 = infinity? substituting the value of x for the expression 1/x?

Answer 3

and question D |x|/x= 0. as O is neither negative nor positive. however if | any number| = positive but in zero it does not make any sense. so it will be 0.

Answer 4

For (a), (b), and (c): Try viewing the graph of y = 1/x. It gives you a good idea of the tendencies of a limit to view its graph. Then it is only a matter of "what happens as x gets closer to 0 coming from the right (positives)? the negatives?"

Answer 5

To say that a limit exists means to have the left-hand and right-hand limits equal, then, for (c).

Answer 6

For (d), I would look at it in cases.
|x| is x when x>=0, and -x when x<0 (because - (-x) = +x)

So on the interval, (0, infinity), |x|/x = +x/x = +1. On the interval (-infinity, 0), you have |x|/x = -x/x = -1.
It should be clear that the two sides are not equal. -1 =/= 1.

Answer 7

They would not "meet" in the middle, essentially, it would just jump from -1 to +1 on the graph and a hole at x=0 on either side.

Answer 8

If there is anything that does not make sense or that I can clarify, feel free to ask! :)

Answer 9

I understand! But I think it is required for me to make some sort of substitution to find the exact value?

Answer 10

For which of the problems? I don't think there "should" be a need for it, but if that was what is asked for it I could think of something.

Answer 11

In any of the cases, direct substitution does not work. You end up with forms that are not well defined like 1/0 and 0/0.

Answer 12

Oh yes! Then I understand! Thanks a lot! :D

Answer 13

Ah, great! You're welcome! :)