1.15 Connecting Limits at Infinity and Horizontal Asymptotes

f(x) = x, as x approaches positive infinity, the function approaches positive infinity, and we would write the limit as: lim x->+infinity f(x) = +infinity This function has no horizontal asymptote

g(x) = x^2, as x approaches positive infinity, the function approaches positive infinity, and we would write the limit as: lim x->+infinity g(x) = +infinity This function has no horizontal asymptote

h(x) = x^2-2x-3, as x approaches positive and negative infinity, the function approaches positive and negative infinity respectively, and we would write the limit as: lim x->+infinity h(x) = +infinity, lim x->-infinity h(x) = -infinity This function has no horizontal asymptote

j(x) = x^3+3x^2+3x+1/(x+1), as x approaches positive infinity, the function approaches 1, and we would write the limit as: lim x->+infinity j(x) = 1 As x moves closerto positive infinity, the denominator (x+1) becomes larger and larger, causing the value of the function to approach the constant value of 1. This results in a horizontal asymptote at y = 1.

k(x) = 1/x, as x approaches positive infinity, the function approaches 0, and we would write the limit as: lim x->+infinity k(x) = 0 As x moves closer to positive infinity, the denominator becomes larger and larger, causing the value of the function to approach 0. This results in a horizontal asymptote at y = 0.

l(x) = 1/x^2, as x approaches positive infinity, the function approaches 0, and we would write the limit as: lim x->+infinity l(x) = 0 As x moves closer to positive infinity, the denominator becomes larger and larger, causing the value of the function to approach 0. This results in a horizontal asymptote at y = 0.

m(x) = x^2+x-6/x+2, as x approaches positive infinity and -2, the function approaches 1 and -3 respectively, and we would write the limit as: lim x->+infinity m(x) = 1, lim x->-2 m(x) = -3 As x moves closer to positive infinity and -2, the denominator becomes larger and larger, causing the value of the function to approach 1 and -3 respectively. This results in horizontal asymptotes at y = 1 and y = -3.

n(x) = x^2-5x+6/x^2-4x+4, as x approaches positive and negative infinity, the function approaches 1 and -1 respectively, and we would write the limit as: lim x->+infinity n(x) = 1, lim x->-infinity n(x) = -1 As x moves closer to positive and negative infinity, the denominator becomes larger and larger, causing the value of the function to approach 1 and -1 respectively. This results in horizontal asymptotes at y = 1 and y = -1.

p(x) = x^3-9x+3/x^2+1, as x approaches positive and negative infinity, the function approaches 0, and we would write the limit as: lim x->+infinity p(x) = 0, lim x->-infinity p(x) = 0 As x moves closer to positive and negative infinity, the denominator becomes larger and larger, causing the value of the function to approach 0. This results in horizontal asymptote at y = 0.

q(x) = (x^2-4)/(x^2-16), as x approaches positive and negative infinity, the function approaches 0, and we would write the limit as: lim x->+infinity q(x) = 0, lim x->-infinity q(x) = 0 As x moves closer to positive and negative infinity, the denominator becomes larger and larger, causing the value of the function to approach 0. This results in a horizontal asymptote at y = 0.