Limits

Learning Objectives

 * Having an intuitionistic understanding of Limits of a function.
 * Understanding algebra of limits.
 * Working out limits of various types of functions.
 * Identifying various indeterminate forms.

Theory
Limits : Its surprisingly hard to rigorously define limits but we can deduce an intuitive explanation from various examples. We say that the limit of function $$f(x)$$ as $$x$$ tends to a is L, if $$f(x)$$ approaches (gets closer to) L , as $$x$$ gets closer to a. If such a number does not exist, we say that the limit does not exist.

This is expressed mathematically as $$\lim_{x \to a} f(x) = L $$

Important Point
What is $$\lim_{x \to 2} f(x) $$where $$f(x) = \left\{ \begin{array}{ll} x & \quad x \in \R - [2] \\ 6 & \quad x = 2 \end{array} \right.$$?

Since as $$x$$ approaches 2, $$f(x)$$approaches 2, $$\lim_{x \to 2} f(x) $$= 2.

This shows that $$\lim_{x \to a} f(x) $$need not be equal to $$f(a)$$even when $$f(a)$$ is explicitly defined.

Concept 2 : Limit laws
In this section we will learn the algebra of limit of functions.

Theory
Before delving into more complex theorems, let us establish the basic ones.

Limit of a constant function is same at each point.

i.e. $$ \lim _{x \rightarrow a} c=c $$

Also limit of $$ f(x) = x $$at $$ a $$is $$ a $$.

i.e. $$ \lim _{x \rightarrow a} x=a $$. This makes sense since the function is defined at all points and is continuous.

Now we are prepared to move forward with the algebra of limits.

$$\begin{aligned} &\lim _{x \rightarrow p}(f(x)+g(x))=\lim _{x \rightarrow p} f(x)+\lim _{x \rightarrow p} g(x) \\ &\lim _{x \rightarrow p}(f(x)-g(x))=\lim _{x \rightarrow p} f(x)-\lim _{x \rightarrow p} g(x) \\ &\lim _{x \rightarrow p}(f(x) \cdot g(x))=\lim _{x \rightarrow p} f(x) \cdot \lim _{x \rightarrow p} g(x) \\ &\lim _{x \rightarrow p}(f(x) / g(x))=\lim _{x \rightarrow p} f(x) / \lim _{x \rightarrow p} g(x) \\ &\lim _{x \rightarrow p} \quad f(x)^{g(x)}=\lim _{x \rightarrow p} f(x)^{\lim _{x \rightarrow p} g(x)} \end{aligned}$$

Activity
Intuitively understanding limit laws.

Concept 3 : Working out limits of functions algebraically
Using the limit laws we studied in the last section, we can manipulate functions and try to solve them algebraically instead of using graphs or tables like we did before.