Limits and Continuity

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Introduction

• For a refresher on limits of scalar-valued functions, watch lecture recording 2

Limit Laws

1. $\underset{x\to c}{\lim }(f(x)+g(x))=\underset{x\to c}{\lim }f(x)+\underset{x\to c}{\lim }g(x)$
2. For any constant $a\in \mathbb{R},\underset{x\to c}{\lim }(a\cdot f(x))=a\cdot \underset{x\to c}{\lim }f(x)$
3. $\underset{x\to c}{\lim }(f(x)\cdot g(x))=(\underset{x\to c}{\lim }f(x))\cdot (\underset{x\to c}{\lim }g(x))$
4. If $\underset{x\to c}{\lim }g(x)\ne 0$, then $\underset{x\to c}{\lim }\frac{f(x)}{g(x)}=\frac{{\lim }_{x\to c}f(x)}{{\lim }_{x\to c}g(x)}$

Need to practice applying and using limit laws

One-Sided Limits

• For some functions, approaching $c$ from either side would produce a different limit
  • In this scenario, there is no true limit, however, there is a special notation to write a limit for either side
• We write:
  • $\underset{x\to c^{-}}{\lim }f(x)=L$ if $f(x)$ gets close to $L$ as $x$ gets close to $c$ with $x<c$
  • $\underset{x\to c^{+}}{\lim }f(x)=L$ if $f(x)$ gets close to $L$ as $x$ gets close to $c$ with $x>c$
• $\underset{x\to c}{\lim }f(x)=L$ only exists if the limits from both sides exist and are equal

Limit Taking Techniques

Squeeze Theorem

Read about Squeeze Theorem on slides
Useful for proving the limit of an oscillating function (e.g. a trigonometric function)

• Worked Example:

  

Diverging to $\pm \infty$

• Whenever a function approaches $\pm \infty$ as $x\to a$, we consider that the limit does not exist
  • It can still be denoted as $\underset{x\to a}{\lim }f(x)=\infty$ or $\underset{x\to a}{\lim }f(x)=-\infty$

L’Hospital’s Rule

• Watch lecture recording or read slides

Polar Coordinates

• $(x,y)\to (0,0)$ corresponds $r\to 0$
  • Where $x^2+y^2=r^2$

Limits of Vector-Valued Functions

• Limit of $r(t)$ exists if the limit of $x(t),y(x)$ and $z(t)$ exist
  • ($r(t)=(x(t),y(t),z(t))$)

Bolded variables represent vectors
e.g. $f(\mathbf{x})$ could mean $f(x,y)$

Continuity of Functions

• A function is continuous if its graph has no graphs
• This is represented mathematically by: ${\lim }_{x\to c}f(x)=f(c)$
  i.e. f is continuous at c

Combining Continuous Functions

1. The functions $af(x)+bg(x)(a,b\in R)$ and $f(x)g(x)$ are continuous at $c$
2. If $g(c)\ne 0$, then $\frac{f(x)}{g(x)}$ is continuous at $c$
3. If $h(t)$ is a function of a single variable $t\in R$, defined on the range of $f$ and continuous at $f(c)$, then $h(f(x))$ is continuous at $c$
4. If $t\in I\subseteq R$, and $r(t)$ is a vector-valued function such that $r(t)\in D$ for $t\in I$ and $r(t_0)=c$, and if $r$ is continuous at $t_0$, then the function $f(r(t))$ is continuous at $t_0$