Limits

Strategy §

Notation §

Rules §

1. lim k = k, for any constant a -> k
2. lim x = a x -> a
3. lim[f(x) $\pm$ g(x)] = lim f(x) $\pm$ lim g(x) x -> a x -> a x -> a
4. lim[cf(x)] = c [limf(x)], for any constant c x -> a x -> a
5. lim[f(x)g(x)] = [lim f(x)][lim g(x)] x -> a x -> a x -> a
6. lim $\frac{f(x)}{g(x)}$ = $\frac{limf(x)}{limg(x)}$ provided that lim g(x) $\ne$ 0 x -> a x -> a
7. lim[f(x)]${}^n$ = [lim f(x)]${}^n$, for any number n x -> a x -> a

Methods §

1. Direct Substitution Limit
2. Factoring Method Limit
3. Multiplying by the Conjugate
4. Change of Variable Method

Notation and meaning §

x -> 1${}^{+}$ = Approaching 1 from the right x -> 1${}^{-}$ = Approaching 1 from the left lim = limit of f(x), as approaches c equals L

lim f(x) = 4 x -> 1

As x approaches 1 from left and right it equals 4

Conditions §

If lim f(x) = lim f(x) then limit exists and we write x -> 1${}^{-}$ x -> 1${}^{+}$ lim f(x) = 4 x -> 1

Limit may exist even if the point is undefined (eg: If f(3) = undefined a limit may still exist as we are x -> 3${}^{+}$ and x -> 3${}^{-}$)

We are talking about the values that f(x) takes when x gets closer and closer to 1 and not f(1). In fact, we may talk about the limit of f(x) as x approaches “a” even when f(x) is undefined,

Book definition §

Example §