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

May 7, 2015

Let $f \left(x\right) = \frac{{x}^{2} + x + 4}{{x}^{3} - 2 {x}^{2} + 7}$

${\lim}_{x \rightarrow a} f \left(x\right) = f \left(a\right)$ if $x = a$ does not make the denominator $0$,

That is: if $a$ is not a solution to ${x}^{3} - 2 {x}^{2} + 7 = 0$.
(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 $x \rightarrow c$ 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 ${x}^{3} - 2 {x}^{2} + 7$, so that case will not arise.