I only recently started learning calculus on my own so correct me if i'm wrong.
When using direct substitution and the denominator equals 0, the limit is undefined, just like any fraction is when its denominator equals 0. However, it's limit can still be found through substitution by making f(x)=g(x) when x≠a.
i.e limx→1 (x2-1)/(X-1) can be written as limx→1 (X+1) (canceling out by factorizing the numerator as a difference of squares). Through substitution we then find the limit to be 2.
I also got the idea that when the numerator equals 0 through substitution the limit doesn't exist. i.e for limX→2 (x^2-x+6)/(x-2) no preliminary algebra exists to factor this and make it possible to find the limit through substitution so the limit doesn't exist.
The Attempt at a Solution
limx→2(x+2)/(x^3+8) doesn't appear to exist. However, on my text book it says it's -1/2...
All help is appreciated! Let me know if anything i said is incorrect.