Question

help find indicated limits

Answer 1

@Astrophysics

Answer 2

@SolomonZelman

Answer 3

\(\Large\color{slate}{\displaystyle\lim_{x \rightarrow~ -5}f(x)}\)


where the function is:

\(\LARGE\color{black}{ f(x) = \begin{cases} & x+4,~~~{\large x<-5} \\ & 4-x,~~~{\large x\ge-5} \end{cases} }\)

Answer 4

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.

\(\Large\color{slate}{\displaystyle\lim_{x \rightarrow~ -5^+}f(x)=\lim_{x \rightarrow~ -5^-}f(x)}\)

Answer 5

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

\(\Large\color{slate}{\displaystyle\lim_{x \rightarrow~ -5^+}(4-x) =\lim_{x \rightarrow~ -5^-}(x+4) }\)

Answer 6

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.

Answer 7

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

Answer 8

so it does not exist

Answer 9

what about the other question?

Answer 10

Yes, from the right side the limit is 9 and from the left side the limit is -1.

Does Not exist is correct

Answer 11

I need to switch browsers. please wait

Answer 12

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

Answer 13

will do without the pic for now

Answer 14

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

Answer 15

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.

Answer 16

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