## burhan101 2 years ago simple limit but i am blank

1. burhan101

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

2. wio

Does it exist?

3. burhan101

yes

4. wio

5. burhan101

not really

6. burhan101

i completely forgot how to solve these except for direct substitution

7. wio

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

8. wio

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. burhan101

okay thank you :)

10. burhan101

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

11. wio

Start by simplifying it into a single fraction.

12. burhan101

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

13. burhan101

ohh okay

14. wio

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

15. wio

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

16. wio

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

17. burhan101

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

18. wio

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. wio

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

20. wio

Continuous at $$-3$$ at least.

21. burhan101

what does that mean ?

22. wio

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. burhan101

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.