## anonymous 4 years ago simple limit but i am blank

1. anonymous

$\huge \lim_{x \rightarrow 5} \frac{ 1 }{ x-5 }$

2. anonymous

Does it exist?

3. anonymous

yes

4. anonymous

5. anonymous

not really

6. anonymous

i completely forgot how to solve these except for direct substitution

7. anonymous

Well, when you can't directly substitute, I recommend putting in really close values (e.g. 5.001 and 4.999).

8. anonymous

Make sure it exists first. Then you can use things like squeeze theorem / l'Hospital's rule. There is no sure way of doing limits, only a series of methods.

9. anonymous

okay thank you :)

10. anonymous

how would i solve limits that have fractions over fraction though ?

11. anonymous

Start by simplifying it into a single fraction.

12. anonymous

$\large \lim x \rightarrow -3 \frac{ \frac{ 1 }{ x } +\frac{ 1 }{ 3 }}{x+3}$

13. anonymous

ohh okay

14. anonymous

(a/b)/c = a/(bc) and a/(b/c) = (ac)/b

15. anonymous

Okay you need to add up the $$1/x$$ and $$1/3$$.

16. anonymous

a/b + c/d = (ad + bc)/(bd)

17. anonymous

$\large \lim \rightarrow -3 \frac{ 3+x }{ 3x^2+9x }$

18. anonymous

I don't think you're doing it correctly. $\frac{1}{x} +\frac{1}{3} = \frac{x+3}{3x}$And then $\Large \frac{\frac{x+3}{3x}}{x+3} = \frac{x+3}{3x(x+3)} = \frac{1}{3x}$

19. anonymous

This function becomes continuous once you have manipulated it a bit.

20. anonymous

Continuous at $$-3$$ at least.

21. anonymous

what does that mean ?

22. anonymous

What does continuous mean? It means: 1) $$f(a)$$ is defined 2) $$\lim_{x \to a}f(x)$$ exists and 3) $$\lim_{x \to a}f(x) = f(a)$$ In general, it means you can just plug $$a$$ into $$f(x)$$ and get the answer to the limit.

23. anonymous

ohh thankyou !

24. Kainui

Ask yourself, what is a limit? Really. It's what happens as you approach a number from both sides. So to approach a number, you can plug in numbers close to it, like 4.999 is close to 5 on the left side while 5.00001 is close on the right side. Make sense? Plugging these in and seeing what you approach is really the essence of a limit and understanding that will let you solve any limit problem by simply graphing it when you get stumped.