Advanced High-School Mathematics

246 CHAPTER 5 Series and Differential Equations

|f(x)−L|< .

Notice that in the above definition we stipulate 0<|x−a|< δrather
than just saying|x−a|< δbecause we really don’t care what happens
whenx=a.

In defining limits involving∞only slight modifications are necessary.

Definition.

Limits at∞. Letf be a function defined for allx > N. We say that

We say that thelimitoff(x)isLasxapproaches∞, and write

xlim→∞f(x) = L,

if for any real number  > 0 , there is another real number K

(which in general depends on) such that wheneverx > K then

|f(x)−L|< .

In an entirely similarly way, we may definex→−∞lim f(x) = L.

Limits of∞. Letfbe a function defined in a neighborhood of the real

numbera. We say that thelimitoff(x)isLasxapproaches∞,

and write

xlim→af(x) = ∞,

if for any real numberN, there is another real numberδ > 0 (which

in general depends onN) such that whenever 0 <|x−a|< δthen

|f(x)|> N.

Similarly, one defines

xlim→af(x) = −∞, xlim→∞f(x) = ∞,

and so on.

Occasionally, we need to considerone-sided limits, defined as fol-
lows.