MATH12002 - CALCULUS I
                        §1.6: Limits at Infinity
                         Professor Donald L. White
                        Department of Mathematical Sciences
                              Kent State University
   D.L. White (Kent State University)                                  1 / 13
  Introduction to Limits at Infinity
   Our definition of lim f (x) = L required a and L to be real numbers.
                    x→a
   In this section, we expand the definition to allow a to be infinite (limits at
   infinity) or L to be infinite (infinite limits).
   Wenowconsider limits at infinity.
   Afunction y = f(x) has limit L at infinity if the values of y become
   arbitrarily close to L when x becomes large enough. Our basic definition is:
   Definition
   Let y = f (x) be a function and let L be a number.
   The limit of f as x approaches +∞ is L if y can be made arbitrarily close
   to L by taking x large enough (and positive).
   Wewrite lim f(x) = L.
             x→+∞
   Compare this to the definition of lim f (x) = L.
                                     x→a
   The definition means that the graph of f is very close to the horizontal
   line y = L for large values of x.
   D.L. White (Kent State University)                                       2 / 13
  Introduction to Limits at Infinity
   Most of the functions we study that have finite limits at infinity are
   quotients of functions. To evaluate these limits at infinity, we will use the
   following idea.
   Basic Principle
   If c is a real number and r is any positive rational number, then
                                   lim   c =0.
                                  x→+∞xr
   If c is a real number and r is any positive rational number such that xr is
   defined for x < 0, then                c
                                   lim    r = 0.
                                  x→−∞x
   This is used as in the following examples.
   D.L. White (Kent State University)                                       3 / 13
  Examples
   Example
   Find
                                        2x3 +5
                                lim     3
                              x→+∞7x +4x+3
   if the limit exists.
   Before we solve this problem, notice that 2x3 + 5 and 7x3 + 4x + 3 both
   approach +∞ as x → +∞ and so
                              3               lim (2x3 +5)
                   lim     2x +5      6= x→+∞                  .
                 x→+∞7x3+4x+3              lim (7x3 +4x +3)
                                         x→+∞
   In order to evaluate the limit, we will first use an algebraic manipulation to
   turn this into an expression whose limit is the quotient of the limits of the
   numerator and denominator.
   Wewill then use the Basic Principle to evaluate these limits.
   D.L. White (Kent State University)                                       4 / 13