6 Limits that are infinite in height As the denominator approaches zero,the magnitude of the fraction getsvery large.vertical asymptote at x=0.If the denominator is positive, then the fraction’s overall sign is positive.If the denominator is negative, then the fraction’s overall sign is negative.The overall limit as x approaches zero does not exist,since the left-hand and right-hand limits don’t match.
7 The overall limit as x approaches zero is Example 4:The denominator is positive after squaring x, so the limit has the same sign from left and right.The overall limit as x approaches zero is“positive without bound,” since the left-hand and right-handlimits both trend toward positive infinity.
8 End Behavior ModelsAn end behavior model describes the height behavior of a function as x approaches positive infinity oras x approaches negative infinity.A function g(x) is:a right-end-behavior model for f if and only ifa left-end-behavior model for f if and only if
9 Find right- and left-end models for Example 7:Find right- and left-end models forAs , approaches zero...whereas x is furtherfrom the x-axis, so x dominates on the right.becomes a right-end behavior model.Test ofmodelAs , is much further away from the x-axis,therefore is dominant on the left.becomes a left-end behavior model.Test ofmodelOur modelis correct!
10 becomes a right-end behavior model. Example 7:becomes a right-end behavior model.becomes a left-end behavior model.On your calculator, compare which pairs of graphsmatch the best on the right and on the left of the y-axis:Window:
11 Example:For rational functions, the end behavior model comes from the highest power termsin the numerator and denominator:f(x) ≈when |x| largedominant terms in numerator and denominator
12 You can adjust the direction of limits: “outbound” limit for an inputequals“inbound” limit (toward x = 0) for the input’s reciprocalp