2.2 Limits at Infinity: Horizontal Asymptotes Infinite Limits: Vertical Asymptotes

Limits when x Increases Without Bound Let f be a function defined on some interval (a, ∞). Then Means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large.

Formal Definition

Limits when x Decreases Without Bound Let f be a function defined on some interval (-∞,a). Then Means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large negative.

Formal Definition

Horizontal Asymptote The line y = b is called a horizontal asymptote of the curve y = f(x) if either or

Example Solution

Crossing Horizontal Asymptotes Unlike vertical asymptotes that can never be crossed, horizontal asymptotes are about behavior for LARGE x values. Behavior close to the origin may cross the horizontal asymptote. What is important is what happens as x gets very large in the positive or negative direction.

Multiple Horizontal Asymptotes If the limits are different depending on the direction, then there may be more than one asymptote – One asymptote will occur as x∞ Another may occur as x - ∞ An example would be:

Limit Laws The limit laws used for definite numbers can also be used for finding limits as x±∞.

Infinite Limit of 1/x It is important to note that and

Infinite Limit of when r > 0 If r > 0 is a rational number, then If r > 0 is a rational number such that is defined for all x, then

Agenda – Monday, August 13th Finding limits analytically hw quiz Finish notes on limits at infinity & horizontal asymptotes 2.1: 8,10,12,14,16,18,20,22,24,26,28,31,32, 33,34,35,36,37,38,39,40,41,42,43,44,49,50,51,54,58,63 due 08/14

Finding Horizontal Asymptotes of Rational Functions Divide both numerator and denominator by the variable with the highest exponent in denominator. NOTE: If you must deal with a radical, remember that when you divide inside the radical you are dividing by the highest exponent times the index Simplify. Take the limit of each term in the numerator and denominator.

Example Solution Find the highest degree n of the terms, then multiply the numerator and the denominator by

Simplify Take limit of each term

Example Solution

Example Solution

Example

Beware of Indeterminate Forms Indeterminate forms involve ∞’s and 0’s from which no conclusion can be drawn. Examples are: ∞-∞ ∞/∞ 0●∞ ∞^0 1^∞ 0/0 0^0

Going Beyond the Indeterminate Form When encountering an indeterminate form, it is necessary to try “trickery” to find the limit. Try “tricks” learned earlier when finding limits such as rationalization, reduction, squeeze theorem, etc. Ask: Is either term increasing more quickly? Then that term will dominate the answer.

Using Asymptotes and Intercepts to Sketch Graphs Find the vertical asymptote and draw that with a dotted line. Label. Find the Horizontal asymptote and draw that with a dotted line. Label. Find the x-intercepts using algebra (factoring, quadratic formula etc.). Plot these. Remember your rules of multiplicity: If the intercept factor is raised to a even power it touches x and returns, If it is raised to an odd power, it passes through.

Vertical Asymptotes: Infinite Limits When the limit moves unchecked to either + or - ∞, the limit does not exist. However, we will use + or - ∞ to denote the function is increasing (decreasing) without bound in either the + or - direction.

What happens as x approaches 0? As x approaches 0 from the left, the equation values increase without bound As x approaches 0 from the right, the equation values increase without bound.

Example

Vertical Asymptote If the function increases without bound toward either + or - ∞ when x approaches a from either the right or left–hand side, the imaginary line at x = a is a vertical asymptote.

Does this function have a vertical asymptote? Yes, at x =0

Does this function have a vertical asymptote? Yes, at x =0

Example Solution So x=2 is definitely a vertical asymptote

Final Thought It is important to remember that infinite limits are not true limits, they only give an indication of a function’s behavior.