Question

Find the following limits, if they exist. Part (g) and (J)

Answer 1

All.

Answer 2

from question #4

Answer 3

g) is quite easy, just substitute x = 6

Answer 4

don't you have prove each sided limit to prove the limit

Answer 5

Well if you can, but if you know that the function is continuous at x=6, you don't need to compute both sides

Answer 6

for j) if you plug in 0, you get an indeterminate form.. so try finding the left and right-sided limits
Consider \(x\rightarrow 0^+\), then x is positive, so \(|x|=x\)
\[ \Large \lim_{x\rightarrow 0^+}\frac{|x|}{x}=\frac{x}{x}=1\]
Consider \(x\rightarrow 0^-\), then x is negative, so \(|x|=-x\)
\[ \Large \lim_{x\rightarrow 0^+}\frac{|x|}{x}=\frac{-x}{x}=-1\]
So the 2-sided limit doesn't exist

Answer 7

sorry the 2nd limit should say \(\large \lim_{x \rightarrow 0^-}\)

Answer 8

thanks for your help. two questions how do you know that a function is continuous at x=6. second question: why do we use 2 different methods for solving essentially similar questions?

Answer 9

Well if you know that f(x)=|x| is continuous (the graph looks like a V ), and g(x)=x-6 is is continuous (a linear function is continuous), then the composition of 2 continuous functions is continuous, i.e f(g(x)) = |x-6|

Answer 10

Usually though you only need to consider right and left sided limits if when plugging x=.. directly gives some undeterminate form or if you have some piecewise function