# Chapter 8: Continuation of Limits

## Limits

We briefly recall the notion of a limit.

﻿$\lim_{x \rightarrow C} f(x) = L \Leftrightarrow$﻿ ﻿$f(x)$﻿ is close to ﻿$L$﻿ when ﻿$x$﻿ is close to ﻿$C$﻿

### Example

Calculate ﻿$\lim_{x \rightarrow 2} (-x^2+5x-2)$﻿

Because the limit is a simple polynomial we may evaluate by substitution.

﻿$\lim_{x \rightarrow 2} (-x^2+5x-2) = -(2)^2+5 \cdot 2 -2$﻿

﻿$=-4 +10 -2 = 4$﻿

### Example

Calculate ﻿$\lim_{z \rightarrow 4} \sqrt{z^2-10}$﻿

﻿$= \sqrt{4^2-10} = \sqrt{16-10}=\sqrt{6}$﻿

We need to be careful with direct evaluation.

### Example

Calculate ﻿$\lim_{x \rightarrow -3} \frac{x+3}{x^2+4x+3}$﻿

﻿$(-3)+3=0$﻿ and ﻿$(-3)^2+4(-3)+3=0$﻿

Thus we get "﻿$\frac{0}{0}$﻿" which is not a number

﻿$\lim_{x \rightarrow -3} \frac{x+3}{x^2+4x+3}$﻿ ﻿$=\lim_{x \rightarrow -3} \frac{x+3}{(x+4)(x+3)}$﻿

﻿$= \lim_{x \rightarrow -3} \frac{1}{x+4}$﻿

﻿$=\frac{1}{-3+4}$﻿

﻿$=-\frac{1}{2}$﻿

## One Sided Limits

### Discuss

What is the behaviour of ﻿$f(x)=\sqrt{x}$﻿ "close to" ﻿$x=0$﻿?

Consider the graph: 1. Sloping up when ﻿$x \geq 0$﻿
2. Not defined for ﻿$x<0$﻿

### Discuss

What is the behaviours of ﻿$f(x) = \frac{|x|}{x}$﻿ "close tp" ﻿$x=0$﻿?

Consider the graph: 1. ﻿$f(x)=1$﻿ when ﻿$x \geq 0$﻿
2. ﻿$f(x)=-1$﻿ when ﻿$x<0$﻿

We observe that there can be very distinct behaviour "on this left" and "on the right".

### Definition

The right hand limit as ﻿$x$﻿ approaches ﻿$x=c$﻿ of ﻿$f(x)$﻿ is

﻿$\lim_{x \rightarrow C^+} f(x) = L$﻿ The left hand limit is

﻿$\lim_{x \rightarrow C^-} f(x) = L$﻿

We obtain ﻿$\lim_{x \rightarrow 0^+} \frac{|x|}{x}=1$﻿ and ﻿$\lim_{x \rightarrow 0^-} \frac{|x|}{x}=-1$﻿

### Example

Consider the following graph: Which of the following exist (are defined)?

﻿$\lim_{x \rightarrow -1^+} f(x)$﻿ ﻿$\lim_{x \rightarrow -1^-} f(x)$﻿

﻿$\lim_{x \rightarrow 3^+} f(x)$﻿ ﻿$\lim_{x \rightarrow 3^-} f(x)$﻿

﻿$\lim_{x \rightarrow 1^-} f(x)$﻿ ﻿$\lim_{x \rightarrow 1^+} f(x)$﻿

### Definition:

The limit ﻿$\lim_{x \rightarrow C} f(x)$﻿ EXISTS if ﻿$\lim_{x \rightarrow c^-} f(x) = \lim_{x \rightarrow c^+} f(x)$﻿

### Example

Pick a value ﻿$A$﻿ so that :

﻿$\lim_{x \rightarrow} f(x)$﻿ exists when ﻿$f(x)=\left\{\begin{matrix} 3x+2 & x \geq 2\\ \ x^2+A& x<2 \end{matrix}\right.$﻿

We need ﻿$\lim_{x \rightarrow 2^-} f(x) = \lim_{x \rightarrow 2^+} f(x)$﻿

﻿$\lim_{x \rightarrow 2^-} f(x) = \lim_{x \rightarrow 2^-} x^2 +A$﻿ ﻿$\Rightarrow$﻿ ﻿$x \rightarrow 2^-$﻿ gives

﻿$=4+A$﻿

﻿$\lim_{x \rightarrow 2^+}=\lim_{x \rightarrow 2^+} 3x+2$﻿ ﻿$\Rightarrow$﻿ ﻿$x \rightarrow 2^+$﻿ gives ﻿$x \geq 2$﻿

﻿$=8$﻿

Thus, ﻿$8=4+A$﻿ and ﻿$A=4$﻿