Question

What is the limit of `(x^2 + x + 4)/(x^3 - 2x^2 + 7)` as x approaches a and when does the limit exist?

Answer 1

Let `f(x) = (x^2 + x + 4)/(x^3 - 2x^2 + 7)`

`lim_(xrarra) f(x) = f(a)` if `x=a` does not make the denominator `0`,

That is: if `a` is not a solution to `x^3 - 2x^2 + 7 = 0`.
(There is one negative and no positive solutions. Though it takes some algebra to see that.)

Additionally, if the numerator and denominator have any common real zeros, (say `c`) then the limit as `xrarrc` will exist, but we'll have to simplify the expression to find the limit.

`x^2+x+4` has no real zeros and is not a factor of `x^3- 2x^2 + 7`, so that case will not arise.