Find the limits of the following functions, noting the role that infinity plays. (a) 2 lim -

Find the limits of the following functions, noting the role that infinity plays.

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(a) 2 lim - x0x (x = 0) As seen in Fig. 3-2, as x approaches zero from the right [x 0+], f(x) approaches positive infinity; as x approaches zero from the left [x 0], f(x) approaches negative infinity. If a limit approaches either positive or negative infinity, the limit does not exist and is written 2 lim x-0+ x = 2 lim = x-0+x The limit does not exist. (b) 2 lim 2 lim 8818X As also seen in Fig. 3-2, as x approaches , f(x) approaches 0; as x approaches -, f(x) also approaches 0. The limit exists in both cases since zero is a legitimate limit, and is written 4x-9x (c) lim 512 34 2 lim == 0 xxx 2 lim = 0 x8-1x As xx, both numerator and denominator become infinite, leaving matters unclear. A trick in such circumstances is to divide all terms by the highest power of x which appears in the function. Here dividing all terms by x leaves lim = lim 4-(9/x) 4-(0) = 4x-9x xx 5.x-34 x-5-(34/x2) 5-(0) 415