Michael Liu
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Differential Calculus

Limits

Definition of Limits

Suppose f:R→Rf : \mathbb R \to \mathbb R is a function defined on some interval containing x=a∈Rx = a \in \mathbb R (except possibly at x=ax=a). We say that the limit of ff as xx approaches aa is L∈RL \in \mathbb R and write

lim⁡x→af(x)=L\lim_{x \to a} f(x) = L

if for every ϵ∈R+\epsilon \in \mathbb R^+ there exists some δ∈R+\delta \in \mathbb R^+ such that

0<∣x−a∣<δimplies∣f(x)−L∣<ϵ0 < \abs{x - a} < \delta \qquad\text{implies}\qquad \abs{f(x) - L} < \epsilon

If there is no L∈RL \in \mathbb R such that the above holds, then we say the limit does not exist, and write

lim⁡x→af(x)=DNE \lim_{x \to a} f(x) = \text{DNE}

We can imagine ϵ\epsilon to represent some amount of maximum change in the vertical axis of the function ff and δ\delta to represent an amount of change in the horizontal axis. Given ϵ\epsilon, we must find some small enough interval around aa such that f(x)f(x) does not deviate from LL by more than ϵ\epsilon.

If the function is "smooth", then picking smaller and smaller δ\delta's makes the vertical deviation away from LL smaller and smaller in that interval. Eventually, we will find a δ\delta that satisfies the condition in the definition.

However, suppose ff has a jump discontinuity at aa. Then in any interval around aa, and for any constant LL, the function deviates at least half the amount of the jump in the function from LL. This in turn implies that such functions do not have a limit at aa.

Consider a function f:R→Rf : \mathbb R \to \mathbb R that is 1 everywhere except for at x=0x = 0, where it is 22.

f(x)={2x=01otherwise f(x) = \begin{cases}2 & x = 0 \\ 1 & \text{otherwise}\end{cases}

We show that

lim⁡x→0f(x)=1 \lim_{x \to 0} f(x) = 1

First, consider an arbitrary ϵ∈R+\epsilon \in \mathbb R^+. We pick δ=1\delta = 1. Note that for any x∈(−δ,δ)x \in (-\delta, \delta) other than x=0x = 0, we have f(x)=1f(x) = 1. Note that ∣f(x)−1∣=0<ϵ\abs{f(x) - 1} = 0 < \epsilon as required.

The definition of the limit given above is rather restrictive – it only permits functions mapping real numbers to real numbers. However, limits appear in many other areas of mathematics involving functions mapping between all sorts of spaces. In fact, some spaces may not have a convenient distance operator that can take the place of subtraction in R\mathbb R. In these cases, different definitions of the limit are required.

Definition of Continuity

A function f:R→Rf : \mathbb R \to \mathbb R is continuous at a point a∈Ra \in \mathbb R if

lim⁡x→af(x)=f(a) \lim_{x \to a} f(x) = f(a)

A function is continuous in an open interval (a,b)(a, b) if it is continous at all points in (a,b)(a, b)

A function is continuous everywhere if it is continous at all points in its domain.

Properties of Limits

Suppose f,g:R→Rf, g : \mathbb R \to \mathbb R are functions defined in a region around a∈Ra \in \mathbb R. Further let lim⁡x→af(x)=Lf\lim_{x \to a} f(x) = L_f and lim⁡x→ag(x)=Lg\lim_{x \to a} g(x) = L_g. The following properties hold.

lim⁡x→ac⋅f(x)= c⋅lim⁡x→af(x)lim⁡x→a[f(x)+g(x)]=lim⁡x→af(x)+lim⁡x→ag(x)lim⁡x→a[f(x)⋅g(x)]=lim⁡x→af(x)⋅lim⁡x→ag(x)lim⁡x→afb(x)=[lim⁡x→af(x)]b\begin{align} &\lim_{x \to a}c \cdot f(x) &=& \, c \cdot \lim_{x \to a} f(x) \\ &\lim_{x \to a} \left[f(x) + g(x)\right] &=& \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \\ &\lim_{x \to a} \left[f(x) \cdot g(x)\right] &=& \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) \\ &\lim_{x \to a} f^b(x) &=& \left[\lim_{x \to a} f(x)\right]^b \end{align}

If p(x)p(x) is a polynomial, then

lim⁡x→ap(x)=p(a) \lim_{x \to a} p(x) = p(a)

Lemma: Constant functions f(x)=cf(x) = c are continuous everywhere

Proof: Let a∈Ra \in \mathbb R and ϵ∈R+\epsilon \in \mathbb R^+ be arbitrary. Pick δ=1\delta = 1 and L=cL = c. On the interval (a−δ,a+δ)(a - \delta , a + \delta), we have f(x)=c=Lf(x) = c = L as required.

Lemma: Linear functions f(x)=mx+bf(x) = mx + b for m,b∈Rm, b \in \mathbb R are continuous everywhere.

Proof: Let a∈Ra \in \mathbb R and ϵ∈R+\epsilon \in \mathbb R^+ be arbitrary. Pick δ=ϵ2m\delta = \frac{\epsilon}{2m} and L=ma+bL = ma+b. On the interval (a−δ,a+δ)(a - \delta, a + \delta), we have f(x)=f(a)+m(x−a)f(x) = f(a) + m(x-a). Then ∣f(x)−L∣=∣m(x−a)∣≤∣mδ∣≤ϵ2<ϵ\abs{f(x) - L} = \abs{m(x-a)} \leq \abs{m\delta} \leq \frac{ \epsilon}2 < \epsilon as required.

Proof of Corollary: We will prove by induction on the degree of the polynomial.

lim⁡x→ac=clim⁡x→ax=alim⁡x→a3x2+2x=3a2+2a\lim_{x \to a} c = c \qquad \lim_{x \to a} x = a \qquad \lim_{x \to a} 3x^2 + 2x = 3a^2 + 2a

Derivatives

Definition of the Derivative

Let f:R→Rf : \mathbb R \to \mathbb R be a function. For a point x∈Rx \in \mathbb R, we define the derivative of ff at xx to be

f′(x)=lim⁡h→0f(x+h)−f(x)h f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

where the limit exists. This defines a function f′:R→Rf' : \mathbb R \to \mathbb R whose domain consists of the points x∈Rx \in \mathbb R where the above limit exists.