Calculus I Lesson 3: More on Limits

1. Warm Up
   1. Graph
2. Questions from DeltaMath?
3. Recall
4. Computing Limits
   1. Direct Substitution
   2. Algebraic Manipulation
   3. Exercise
5. Conjugates
   1. Exercise
   2. Example
6. Wavy
   1. Graph
7. Upcoming

Warm Up

1. Graph $f(x) = \frac{x^3-1}{x^2-1}$. (Graphing calculator / Desmos)
2. For the points where $f(x)$ is not defined, does the graph of $f$ appear to approach a particular (finite) value?
3. Use the distributive property to show $x^3 - 1 = (x-1)(x^2+x+1)$ and $x^2 - 1 = (x-1)(x+1)$.
4. Simplify $\frac{(x-1)(x^2+x+1)}{(x-1)(x+1)}$.

Graph

• Notice that ${\displaystyle \lim_{x \rightarrow 1}} f(x) = \frac{3}{2}$. But $f(1)$ is not defined?
• Whenever $x \neq 1$, $f(x) = \frac{x^2+x+1}{x+1}$. So, as $x \rightarrow 1$, $f(x) \rightarrow \frac{1+1+1}{1+1} = \frac{3}{2}$.
• What happens as $x \rightarrow -1$?
  • Numerator $\rightarrow -2$, denominator $\rightarrow 0$.
  • Nothing cancels.
  • Hence: asymptote! ($\frac{-2}{.000001}$ is a large negative number)

Questions from DeltaMath?

Recall

• ${\displaystyle\lim_{x\rightarrow a^-}} f(x)$ refers to the limit as $x \rightarrow a$ from the left.
• ${\displaystyle\lim_{x\rightarrow a^+}} f(x)$ refers to the limit as $x \rightarrow a$ from the right.
• If both of the above exist and are equal, then ${\displaystyle \lim_{x\rightarrow a}}f(x)$ exists, and is equal to them both.

Computing Limits

3 ways to compute limits:

1. Direct substitution
2. Algebraic manipulation
3. Squeeze Theorem

Eventually: 4th way (l’Hôpital’s rule). If you know that: don’t use it yet.

Direct Substitution

Easiest method. Works for “nicely behaved functions”:

• Polynomials
• Rational functions, if the denominator is not 0
• sin, cos, tan (if defined)
• exponential, logarithmic functions (if defined)

These functions are continuous.

Algebraic Manipulation

• Can we cancel something?
• Can we multiply by a conjugate?

In other words: can we make our function equivalent to one of the ones where we can use direct substitution, except possibly at one, isolated point? (Ex: warm-up)

(Later: squeeze theorem.)

Exercise

Use algebraic manipulation to compute these limits:

• ${\displaystyle \lim_{x\rightarrow 2}} \frac{x^2-4x+4}{x^2-4}$
• ${\displaystyle \lim_{x\rightarrow 0}} \frac{x}{x^3}$

Conjugates

${\displaystyle \lim_{x \rightarrow 4}}\frac{\sqrt{x} - 2}{x -4}$

Direct substitution? $\frac{2-2}{4-4}$ Indeterminate, direct substitution fails!

Use algebra: multiply by the conjugate of $\sqrt{x}-2$:

\[\begin{align} &(\frac{\sqrt{x}-2}{x-4})(\frac{\sqrt{x}+2}{\sqrt{x}+2}) \\ = &\frac{x-4}{(x-4)(\sqrt{x}+2)} \\ = &\frac{1}{\sqrt{x}+2}. \end{align}\]

So as $x \rightarrow 4$, $\frac{\sqrt{x}-2}{x-4} \rightarrow \frac{1}{4}$.

Exercise

Compute ${\displaystyle \lim_{x\rightarrow 9}\frac{\sqrt{x}-3}{x-9}}$.

Example

Compute ${\displaystyle \lim_{x\rightarrow 0}\frac{\sqrt{x+1}-1}{x}}$.

Direct substitution results in $\frac{0}{0}$. Indeterminate. So multiply by conjugate:

\[\begin{align} (\frac{\sqrt{x+1}-1}{x})(\frac{\sqrt{x+1}+1}{\sqrt{x+1}+1}) \\ = \frac{x+1-1}{x(\sqrt{x+1}+1)} \\ = \frac{x}{x(\sqrt{x+1}+1)} \\ = \frac{1}{\sqrt{x+1}+1}. \end{align}\]

As $x \rightarrow 0$, this expression $\rightarrow \frac{1}{2}$

Wavy

Compute ${\displaystyle \lim_{x\rightarrow 0}} \sin(\frac{1}{x})$. What does this look like as $x \rightarrow 0$?

• Can pick small values of $x$: $\sin(\frac{1}{.00001})$?
• Smarter values of $x$ to use? I know what $\sin(\pi/2)$ is.
• What $x$ value is that, for $y = \sin(\frac{1}{x})$?

Use reciprocals of ones we know. So: $x = \frac{2}{\pi}$, $\frac{2}{99999\pi}$, etc.

Graph

What is going on?

Upcoming

• DeltaMath: Limits due tonight
• Textbook homework: due Monday, 9/15:
  • Section 1.4 #216
  • Section 2.2 #36, 38, 46, 48, 76
  • Section 2.3 #84, 96, 128
  • Write your answers neatly on paper, with your name and Calc I HW 3 written on the top. Staple all the pages together! Bring this in to class on Monday!
  • We will go over some of these problems this Thursday! Be prepared to present some of your work / ask questions on problems you are confused on!
• DeltaMath: More Limits due Thursday, 9/19.
• Thursday: asynchronous lesson + small groups.
  • We can work through some of the homework problems (come prepared with questions about them!), DeltaMath, etc.
  • Can work through examples done in class or from the notes.