 Limit  the expected / intended value of a function  A limit can involve ∞ in two ways:  You can expect a limit to be equal to ±∞ (vertical asymptote, limit DNE)  You can expect a value of a function as x approaches ±∞ (horizontal asymptote)

 As x approaches 3, the value of this function approaches ∞… the limit does not exist

RHL ≠ LHL; neither limit has a defined value From right From left

 When you cannot “get rid of” division by zero in a limit, this represents a vertical asymptote  The limit approaches ±∞  The limit does not exist

 As x gets larger and larger (approaches ∞) what is the intended value of this function?

 If no horizontal asymptote exists

 To evaluate without graphing, you compare the largest degree of the denominator to the largest degree of the numerator

 As x gets larger, denominator > numerator, smaller # / larger # gives smaller and smaller quotients… the overall result will tend towards zero

 As x gets larger, numerator > denominator, larger # / smaller # gives larger and larger results… the overall results is that the value of the function continues to grow and there will be no asymptote

 As x gets larger, the numerator and denominator grow in proportion to each other and the effect of lower-degree terms becomes less  Value of the function as x approaches ±∞ equals the ratio of the coefficients of the largest degrees

 There is no horizontal asymptote; the limit does not exist

 Graphically there is a horizontal asymptote, can also use Sandwich / Squeeze Theorem

 Rules apply for approaching +∞ or - ∞  Often there is little, if any work to show  Graph and record answer in blank provided  Circle relevant exponents and record answer  Watch out for questions involving several fractions  combine to a single fraction first