let h be the function defined by h(x) = (7x²+14x)/(x² - 4). answer the following questions. a) at what x…

let h be the function defined by h(x) = (7x²+14x)/(x² - 4). answer the following questions. a) at what x - value is there a removable discontinuity? x = - 2 b) what is the vertical asymptote? x = 2 c) lim(x→ - 2) h(x) = - 7

let h be the function defined by h(x) = (7x²+14x)/(x² - 4). answer the following questions. a) at what x - value is there a removable discontinuity? x = - 2 b) what is the vertical asymptote? x = 2 c) lim(x→ - 2) h(x) = - 7

Answer

Explanation:

Step1: Factor the function

Given (h(x)=\frac{7x^{2}+14x}{x^{2}-4}), factor the numerator and denominator. Numerator: (7x^{2}+14x = 7x(x + 2)), denominator: (x^{2}-4=(x + 2)(x - 2)). So (h(x)=\frac{7x(x + 2)}{(x + 2)(x - 2)}), (x\neq\pm2).

Step2: Find removable discontinuity

A removable discontinuity occurs when a factor in the numerator and denominator cancels out. Here, the factor ((x + 2)) cancels. The (x) - value for the removable discontinuity is (x=-2) since when (x=-2), the original function is undefined but the simplified - function (after canceling) is well - defined.

Step3: Find vertical asymptote

A vertical asymptote occurs at the (x) - values that make the denominator of the simplified non - canceled function equal to zero. Set (x-2 = 0), then (x = 2).

Answer:

The (x) - value for the removable discontinuity is (x=-2) and the vertical asymptote is (x = 2).