Page 63 - Calculus Workbook For Dummies

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Chapter 4: Nitty-Gritty Limit Problems

1 2 1
15. Evaluate lim x sin 2 n. 16. Evaluate lim x cos x m.
c
d
x " 0 x x " 0
Solve It Solve It

Into the Great Beyond: Limits at Infinity

To find a limit at infinity lim or lim l, you can use the same techniques from the
b
x " 3 x " - 3
bulleted list in the “Solving Limits with Algebra” section of this chapter in order to
change the limit expression so that you can plug in and solve.
If you’re taking the limit at infinity of a rational function (which is one polynomial
2
3 x - 8 x + 12
divided by another, such as 3 2 ), the limit will be the same as the y-value
5 x + 4 x - - 2
x
of the function’s horizontal asymptote, which is an imaginary line that a curve gets closer
and closer to as it goes right, left, up, or down toward infinity or negative infinity. Here
are the two cases where this works:
Case 1: If the degree of the polynomial in the numerator is less than the degree of
the polynomial in the denominator, there’s a horizontal asymptote at y = 0 and
the limit as x approaches 3 or 3- is 0 as well.
Case 2: If the degrees of the two polynomials are equal, there’s a horizontal
asymptote at the number you get when you divide the coefficient of the highest
power term in the numerator by the coefficient of the highest power term in the
denominator. This number is the answer to the limit as x approaches infinity or
negative infinity. By the way, if the degree of the numerator is greater than the
degree of the denominator, there’s no horizontal asymptote and no limit.