## anonymous one year ago help find indicated limits

1. anonymous

2. anonymous

3. anonymous

@Astrophysics

4. anonymous

@SolomonZelman

5. SolomonZelman

$$\Large\color{slate}{\displaystyle\lim_{x \rightarrow~ -5}f(x)}$$ where the function is: $$\LARGE\color{black}{ f(x) = \begin{cases} & x+4,~~~{\large x<-5} \\ & 4-x,~~~{\large x\ge-5} \end{cases} }$$

6. SolomonZelman

So, for a two-sided limit $$\Large\color{slate}{\displaystyle\lim_{x \rightarrow~ -5}f(x)}$$ to exist, the limit from the right side and the limit from the left side have to be equal to each other. $$\Large\color{slate}{\displaystyle\lim_{x \rightarrow~ -5^+}f(x)=\lim_{x \rightarrow~ -5^-}f(x)}$$

7. SolomonZelman

What is our function from the left of -5? (when x is less than -5) it is: x+4 and the function from the right side is 4-x. So this limit: $$\Large\color{slate}{\displaystyle\lim_{x \rightarrow~ -5^+}f(x)=\lim_{x \rightarrow~ -5^-}f(x)}$$ would go the following way $$\Large\color{slate}{\displaystyle\lim_{x \rightarrow~ -5^+}(4-x) =\lim_{x \rightarrow~ -5^-}(x+4) }$$

8. SolomonZelman

if both sides are not equivalent to each other, then the two sides limit Does Not Exist (DNE). if both sides are equivalent, then this value they are both equal is going to be your answer.

9. SolomonZelman

now, do the substitution directly (plug in -5 for x on both sides)

10. anonymous

so it does not exist

11. anonymous

12. SolomonZelman

Yes, from the right side the limit is 9 and from the left side the limit is -1. Does Not exist is correct

13. SolomonZelman

I need to switch browsers. please wait

14. SolomonZelman

I will draw something I can't use on this browser now...

15. SolomonZelman

will do without the pic for now

16. SolomonZelman

Do you see the part of this (piece wise) function that goes from the right to the point where x=3?

17. princeharryyy

the function is a broken line at point 3, on x-axis with the lines being with different slope and a point at the top the limit simply doesn't exist.

18. SolomonZelman

Em, do you see that segment where the line goes from the right to the point x=3?