Evaluate lim x->2 (x2 + 2x - 12)/(x2 - 2x)

The limit to evaluate is:

Substitute $x=2$ into the expression:
Numerator: $2^2 + 2(2) - 12 = 4 + 4 - 12 = -4$
Denominator: $2^2 - 2(2) = 4 - 4 = 0$
Since direct substitution results in $\frac{-4}{0}$, the limit does not exist in the form of a finite number. It approaches either $\infty$ or $-\infty$.

Let $f(x) = \frac{x^2 + 2x - 12}{x^2 - 2x}$.
As $x \to 2$, the numerator approaches $-4$.
The denominator $x^2 - 2x = x(x-2)$.
As $x \to 2^+$, $x-2$ is a small positive number, so $x(x-2)$ is a small positive number. Thus, $f(x) \to \frac{-4}{\text{small positive}} \to -\infty$.
As $x \to 2^-$, $x-2$ is a small negative number, so $x(x-2)$ is a small negative number. Thus, $f(x) \to \frac{-4}{\text{small negative}} \to +\infty$.
Since the left-hand limit and the right-hand limit are not equal, the limit does not exist.

The limit is:

The function is $f(x) = \frac{x^2 + 2x - 12}{x^2 - 2x}$. We need to check its continuity at $x=27$.

A function $f(x)$ is continuous at a point $x=a$ if three conditions are met:

1. $f(a)$ is defined.
2. $\lim_{x \to a} f(x)$ exists.
3. $\lim_{x \to a} f(x) = f(a)$.

Substitute $x=27$ into the function:

Since the denominator $675 \neq 0$, $f(27)$ is defined.

For a rational function, if the denominator is not zero at the point, the limit can be found by direct substitution.

The limit exists and is equal to $f(27)$.

Since $f(27)$ is defined and $\lim_{x \to 27} f(x) = f(27)$, the function is continuous at $x=27$.

The function is continuous at $x=27$.