1.5 Infinite Limits Main Ideas Determine infinite limits from the left and from the right. Find and sketch the vertical asymptotes of the graph of a function.

Vertical Asymptote Let f and g be continuous on an open interval containing c. If f(c) ≠ 0, g(c) = 0, and there exists an open interval containing c such that g(x) ≠ 0 for all x ≠ c in the interval, then the graph of the function given by h(x) = Has a vertical asymptote at x = c. (Rational function and Trigonometric functions) *Note 0/0 (hole)is an indeterminate form. You need to use a previously discussed analytical method to determine the limit as x ⇾ c. ***Both a hole and an asymptote can exist in the same problem.

Infinite limit A limit in which f(x) increases or decreases without bound as x approaches c. This does not mean the limit exists. It tells you how the limit “fails to exist” by denoting the unbounded behavior of f(x) as x approaches c. One – sided limit ∞ If the function is increase as it approaches c. ∞ If the function is decreasing as it approaches c. Limit ∞ If both sides of the function at c is increasing ∞ If both sides of the function at c is decreasing DNE If one side of the function is increasing while the other side of the function is decreasing.

Graphical approach Table approach Algebraic approach Look for a vertical asymptote Table approach Change in x-values is very small but y-values do not converge upon one number, instead they continue to get larger (∞) or smaller (-∞) Algebraic approach Evaluate using direct substitution and you get #/0. You need to pick at least two values close to c and plug it into the function to determine the type of infinity.