you're finding a one-sided limit, so no problem ... do like the book says.
So the question is (and sorry for butchering this... I have tried Latex but cannot figure it out for the life of me)
limit[x:1-,-3x+2] (the limit on (-3x+2) as x approaches 1 from the left)
(PS. Tell me if this notation is acceptable, or my wording is bad...)
The book we use provides solutions, and the solutions show simply subbing in 1 to the (-3x+2), but that wouldn't work if it was piecewise, right?
So how would you do this other than using values like .999999 ?
Okay, I understadn that as long as the domain is not restricted, that works out fine... But what if it was restricted so that x>1 ..... Then this wouldn't work out, correct? (And to tell you the WHOLE truth, that is what the book does... The question is actually a piecewise, and it just subs in 1 from both sides)
I am sorry, I do not have the book on me anymore. I will likely check with the instructor tomorrow but... I'll see if I remember this all.
essentially it was:
f(x)= {-3x+2}, x<1 and {x^2}, x>=1
The questions asks us to find the limits from both directions, at x=1 ...
Then the SOLUTION shows them subbing 1 into the first side of the piecewise function .... That doesn't seem right to me.... 1 is not in the domain of that question so subbing it into there will not return the correct y value.
ok, so here's the thing. with limits, we don't actually care about being "at" the point, but about being "close" to the point. now is asking, when gets "close" to 1 coming from the left (meaning, the x-values are always a little less than 1 but can get arbitrarily close to 1) what value does get "close" to?
now, your piecewise function is defined as for . so your left hand limit is concerned with this definition of . now, as long as the funciton is continuous, you can just plug in the value. so you plug in 1 into
Now, if they asked for , they are asking for the limit as gets close to one from the right. hence you look at values where is a little bigger than 1. so you look for how is defined for . here, we have for , so that