Answer
2a. Factor the numerator
2b. Expand the numerator
2c. Rationalize the numerator
2d. Combine the 2 fractions in the numerator under a common denominator
Work Step by Step
2a. Since $x^{2}-4$ is equal to $(x+2)(x-2)$, $\displaystyle\lim_{x\to2} \frac{x^2-4}{x-2}$ can now be turned into $\displaystyle\lim_{x\to2} (x+2)$ by cancelling $(x-2)$ from the numerator and denominator.
2b. The ultimate goal is to get rid of the $h$ in the denominator, so we have to expand the numerator to get an expression such that it can be represented as $h$ multiplied by an expression (this allows for simplification).
2c. We want to factor out $x$, but we have a radical on the $x$ in the numerator. Therefore, we need to multiply the numerator by the conjugate to get $x$ in the numerator, thereby allowing the simplification.
2d. In order to simplify this limit, we need "$x-7$" in the numerator, meaning that the two fractions must be combined to create a term that has has $x-7$ (which is $\frac{x-7}{7x}$) and then we can simplify by $x-7$.
