Unit 4 - Differential Calculus
Required Prior Knowledge
Questions
Calculate Consider the function$$f\left(x\right)=\frac{x^{2}+3x-4}{x-2}$$Evaluate \(f\left(3\right)\).
Why must we exclude \(x=2\) from the domain of the function?
Solutions
Get Ready
Questions
Complete this table of values:
| \(x\) | \(0.6\) | \(0.7\) | \(0.8\) | \(0.9\) | \(1\) | \(1.1\) | \(1.2\) | \(1.3\) | \(1.4\) |
|---|---|---|---|---|---|---|---|---|---|
| \(y = \frac{x^2 - 1}{x - 1}\) |
What about when \(x=0.99\)?
Or \(x=0.999\)?
Or \(x=1.0001\)?
What can we say about the value of the function as \(x\) gets closer and closer to \(1\)?
Solutions
Notes
The limit of a function is the value that \(f\left(x\right)\) approaches as \(x\) gets closer to a particular value.
There are three approaches to find a limit:
Numerical - analyse what happens numerically as the value of \(x\) gets closer to the given value. This is easily done with technology
Graphical - analyse what happens in the graph near to the value of \(x\). This is easily done with technology too.
Algebraic - use algebra to simplify the function in some way to determine what happens.
Examples and Your Turns
Example
Find the limit of \(x^{2}\) as \(x\) approaches \(2\).
Example
Find the limit of \(\frac{x^{2}+3x}{x}\) as \(x\) approaches \(0\).
Your Turn
Notes (HL)
There are three possible outcomes when trying to determine a limit:
The limit exists and is equal to the function;
The limit exists but is not equal to the function;
The limit does not exist.
For the limit to exist, the limit from the left must equal the limit from the right.
That is$$\lim_{x \to c} f = \lim_{x \to c^{+}} f = \lim_{x \to c^{-}} f$$Note that \(f\left(c\right)\) does NOT need to be defined for the limit to exist.
Examples and Your Turns (HL)
Example
a) Sketch the graph of $$y=\frac{2^{x}-1}{x}, x\ne 0$$
b) Find \(\lim_{x\to 0} \frac{2^{x}-1}{x}\) giving your answers to 2 decimal places.
Your Turn
a) Sketch the graph of $$y = \begin{cases} x - 3, & x < 2 \\ x + 1, & x > 2 \end{cases}$$
b) Find the value of the function as \(x\) gets closer to \(2\) from both the left and the right.
Your Turn
For each function, find the limit, if it exists.
| # | Function | Answer | # | Function | Answer |
|---|---|---|---|---|---|
| 1. | \(\lim_{x \to -1} \frac{x^2 - 1}{x + 1}\) |
\(-2\)
?
|
2. | \(\lim_{x \to 1} \frac{x^3 - 1}{x - 1}\) |
\(3\)
?
|
| 3. | \(\lim_{x \to 2} \begin{cases} 3x - 1, & x < 2 \\ \frac{1}{x^2 - 1}, & x > 2 \end{cases}\) |
DNE
?
|
4. | \(\lim_{x \to 0} \frac{|x|}{x}\) |
DNE
?
|
| 5. | \(\lim_{x \to 6} (x - 6)^{\frac{2}{3}}\) |
\(0\)
?
|
6. | \(\lim_{x \to 3} \lfloor x \rfloor\) |
DNE
?
|
You can see a brief explanation by hovering your mouse over the revealed answer, and you can reveal the graphs of each function below to see the relationship.
Notes (HL)
A function is continuous at \(x=c\) if $$\lim_{c\to c}f\left(x\right)=f\left(c\right)$$
That is, if it satisfies the following three conditions:
\(f\) is defined at \(c\)
The limit of \(f\) at \(c\) exists (the limit from the right equals the limit from the left)
The limit is equal to the value of \(f\left(c\right)\)
A function is said to be continuous if it is continuous for all values of its domain.
We can see continuity by looking at the graph. If we can draw the graph without removing the pencil from the paper, it is continuous. If we need to remove the pencil then it is discontinuous at that point.
All polynomials and exponential functions are continuous. As are the basic three trigonometric functions.
Examples and Your Turns (HL)
Example
Graph the function$$y=\frac{x^{2}-1}{x-1}$$Determine if the function is continuous.
Your Turn
a) Sketch the graph of$$f\left(x\right) = \begin{cases} 1, & x \le -1 \\ -x, & -1 < x < 0 \\ 1, & x = 0 \\ -x, & 0 < x < 1 \\ 1, & x \ge 1 \end{cases}$$
b) Find the limits, if they exist, as \(x\) approaches \(-1\), \(0\) and \(1\).
c) Determine if \(f\) is continuous as \(x=-1\), \(x=0\) and \(x=1\).
Your Turn
$$f\left(x\right) = \begin{cases} \frac{x^3 - 3x^2 + 4}{x + 1}, & x \neq -1 \\ k, & x = -1 \end{cases}$$Determine the value of \(k\) for \(f\left(x\right)\) to be continuous.
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.
Taking it Deeper
Conceptual Questions to Consider
What do we mean by ‘\(x\) approaches \(2\)’? How is this different to the value of \(x\) at \(2\)?
Why does \(f\left(x\right)=\frac{x^{2}-4}{x-2}\) have a hole at \(x=2\)? Why is it not the same as \(g\left(x\right)=x+2\) even though the fraction simplifies to the same algebraic expression?
Describe how to visually tell if a function is continuous. How does this link to the formal definition of continuity?
Common Mistakes / Misconceptions
Substituting a value for \(x\) too early, usually leading to \(\frac{0}{0}\).
Thinking that the limit at \(x=c\) must be equal to the value of the function at \(x=c\).
Assuming an algebraic method does not work when you end up with \(\frac{0}{0}\) - we will see later how to work with this.
Connecting This to Other Skills
We have used our knowledge of What Functions Are (2.2), Function Notation (2.3) and Graphing Functions (2.4), as well as knowledge of Domains an Ranges (2.6).
The definition of Differentiation from First Principles (4.4) is evaluating a limit. Similarly, integral calculus (5.9) is defined as the limit of an infinite process.
We return to the more formal analysis of limits when we see l’Hopital’s Rule (8.1) which enables us to analyse more difficult limits algebraically.
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?