Questions: Evaluate the limit and discuss vertical asymptotes: lim x→2 (x^2-4)/(x-2)=(x^2-4)/(x-2)=((x-2)(x+2))/(x-2)=(2-2)/0=4/0 Explain your steps and why this function does not have any vertical asymptote at x=2.

Solution Steps

We start with the expression \[ \frac{x^2 - 4}{x - 2}. \] The numerator \(x^2 - 4\) can be factored as \[ (x - 2)(x + 2). \]

Substituting the factored form into the expression gives us \[ \frac{(x - 2)(x + 2)}{x - 2}. \] We can cancel the \((x - 2)\) term in the numerator and the denominator, resulting in \[ x + 2. \]

Now, we evaluate the limit as \(x\) approaches \(2\): \[ \lim_{x \to 2} (x + 2) = 2 + 2 = 4. \] Since the factor \((x - 2)\) was canceled, the function does not have a vertical asymptote at \(x = 2\).

Final Answer