Question

We're reviewing limits from the previous chapter, but I'm still having some difficulty with some problems:

lim tan(pi(x/4)) as x-> -7

lim (x^2 - x^3(cos(1/x))/3x^2 as x-> 0

I know the answer to the first problem is 1 and to the second problem it's 1/3 but I'm not sure how to get these answers. Any help is greatly appreciated! :)

Answer 1

The first step in any limit question is to actually substitute the approach value into the expression.
If you get a real number as an answer, that is it, you are done
so Lim tan(πx/4) as x ---> -7
= lim tan(-7π/4)
= 1

in lim (x^2 - x^3(cos(1/x)/(3x^2) as x ---> 0
= lim (1-xcos(1/x))/3

no matter what cos(1/x) is , xcos(1/x) ---> 0 as x-->0
(multiplying anything by zero, is zero!)

so lim (1-xcos(1/x))/3
= (1 - 0)/3 = 1/3

Answer 2

Reading these answers makes a whole lot more sense after a night's sleep, haha. Thank you so much!