For each of the following, check for discontinuities and state the equation of any vertical asymptotes. Conduc

Question

For each of the following, check for discontinuities and state the equation of any vertical asymptotes. Conduct a limit test to determine the asymptote from above or below.

a) y = x / x + 5

b) f(x) = 2x / x2 - 1

c) g(t) = 3t2 + 4 / t2 - 1

d)y = 3x2 - 8x - 7 / x - 4

Please show your work this would be appreciated.

Answer 1

a) vertical asymptote at x = -5. for x > -5, graph approaches horizontal asymptote y = 1 from below, since x+5 > x, and for x < -5, graph approaches y=1 from above. 5/(5+5) = 1/2 < 1. -10/(-10+5) = 2 > 1.

b) vertical asymptotes when x² = 1, x = ±1. since lim (x→∞) 2x/x² = 0, horizontal asymptote is y = 0 for x > 1 and x < -1.

not clear what you mean by determining the asymptote from above or below.

c) vertical asymptotes at t = ±1. horizontal asymptote at lim (t→∞) 3t²/t² = 3

d) vertical asymptote at x = 4. dividing 3x² - 8x - 7 by x - 4 gives 3x + 4 as the slant asymptote. remainder 9/(x-4) → 0 as x→∞.