## hawkfalcon 3 years ago limits:O

1. hawkfalcon

$\lim_{x \rightarrow 5} \frac{ x-5 }{ \left| x-5 \right| }$

2. vf321

Well, do you know that$\lim_{x\rightarrow a}f(x)=\lim_{x\rightarrow a^+}f(x)\cap\lim_{x\rightarrow a^-}f(x)$I.e., for 1 variable, the bilateral limit exists iff both lateral limits exist and are the same?

3. hawkfalcon

Erm, yes.

4. vf321

Okay. Then coming in from the right at 5, what is your limit equal?$\lim_{x \rightarrow 5^+} \frac{ x-5 }{ \left| x-5 \right| }=?$

5. hawkfalcon

0/0

6. hawkfalcon

oh wait. $infinity$

7. hawkfalcon

would it be DNE?

8. vf321

The right-side limit is not DNE. When solving limits you can only say that$\lim_{x\rightarrow a}f(x)=f(a)$ if f(a) exists.

9. vf321

Think about it. We're approaching the limit from the right. When x = 6, what's the limit? When x = 5.01, what's the limit?

10. hawkfalcon

okay. Wait each side would be infinity right?

11. vf321

Try plugging in values I showed you. Also, a note for the prior statement I made: f has to be continuous.

12. hawkfalcon

at 6 it would be -1/2

13. hawkfalcon

right?:3

14. hawkfalcon

I read that it has to be either DNE, infinity, or negative infinity

15. hawkfalcon

@vf321 sorry, i'm confused :(:#

16. vf321

Lets try this again.$\frac{6-5}{|6-5|}=1$ how on earth did u get 1/2?

17. hawkfalcon

I was accidentally looking at a different problem:3

18. vf321

okay. now try 5.0001

19. hawkfalcon

But okay, then 5.0001 = 1 too.

20. vf321

okay. try 5.000000000000000001

21. hawkfalcon

Anything >5 will be 1

22. vf321

good. In fact, you can say$\lim_{x \rightarrow 5^+} \frac{ x-5 }{ \left| x-5 \right| }=1$

23. vf321

Now try$\lim_{x\rightarrow 5^-}f(x)=?$From the left, on your own.

24. hawkfalcon

o.o okay that makes sense. Let me try...

25. hawkfalcon

-1

26. hawkfalcon

So it's DNE, because they are different.

27. vf321

yes! and, if you were to graph the function on a CAS (or Wolfram|Alpha), you'd see there's a jump discontinuity at 0.

28. hawkfalcon

Do I write that?:O

29. vf321

no but its for ur understanding.

30. hawkfalcon

Oh okay, thank you:)

31. vf321

remeber to close the question.