Evaluate four of the following limits. \[ \lim _{x \rightarrow 2} \frac{x^{4}-4 x^{2}}{x-2} \] (b)

To evaluate the limit \[ \lim _{x \rightarrow 2} \frac{x^{4}-4 x^{2}}{x-2}, \] we first simplify the expression in the numerator \( x^{4} - 4x^{2} \). We can factor this expression. Notice that: \[ x^{4} - 4x^{2} = x^{2}(x^{2} - 4) = x^{2}(x - 2)(x + 2). \] Now we can rewrite the limit: \[ \lim _{x \rightarrow 2} \frac{x^{2}(x - 2)(x + 2)}{x - 2}. \] We can cancel \( (x - 2) \) in the numerator and denominator (noting that we cannot do so at \( x = 2 \), which we are approaching): \[ \lim _{x \rightarrow 2} x^{2}(x + 2). \] Now we can substitute \( x = 2 \): \[ = 2^{2}(2 + 2) = 4 \cdot 4 = 16. \] Thus, the limit is \[ \boxed{16}. \]