2.1 - Intro to Limits

Definition of a limit

• The value that a function approaches as the input approaches some value.
• A limit is different to a function value.
  • Limits: all about approaching a value
    • ${\displaystyle \underset{x\to 2}{lim}f(x)=5}$
      • Pronounced “the limit as x approaches 2 of $f(x)$ is 5”
  • Function values: are evaluated at a point
    • $f(x)=2$
  • Partial limits show what side a limit comes from.
    • ${\displaystyle \underset{x\to 2^{-}}{lim}f(x)=5}$ comes from the left
    • ${\displaystyle \underset{x\to 2^{+}}{lim}f(x)=5}$ comes from the right
    • Partial limits always exist.

Whether or not a limit exists

• For a limit to exist at $x$, the limit from the left side must equal the limit from the right side.
  • In algebraic terms, the ${\displaystyle \underset{x\to c^{-}}{lim}f(x)}$ must equal ${\displaystyle \underset{x\to c^{+}}{lim}f(x)}$.
  • Otherwise the limit does not exist (DNE)
    • Use DNE when the right and left hand limits are not equal to each other. This includes things like vertical asymptotes for things like rational functions.
    • Very rarely, use no limit for things that don’t even have a right/left hand limit.
      Example of a limit not existing: ${\displaystyle \underset{x\to 0}{lim}\frac{1}{x}}$
      

Finding limits algebraically

Strategies:

1. Direct substitution: just plug the limit into the function. Sometimes this doesn’t work because you get $0/0$.
2. Other strategies:
   1. Factoring
   2. Expanding
   3. Rationalizing
   4. Finding a common denominator
   5. Multiplying by a conjugate
3. The big guns - L'Hopital's Rule

Limit of $\sin (x)/x$

An important fact that gets used a lot in these types of problems is the fact that ${\displaystyle \underset{x\to 0}{lim}\frac{\sin (ax)}{ax}=1}$ (where $a$ can be any non-zero constant).

• This can be used in a variety of creative ways.

Average and instantaneous rates of change

• Average rates of change don’t require limits. They are just slope, and slope is calculated with two points.
• Instantaneous rates of change do, because you just have one point to work with.

Example

An object dropped from rest from the top of a tall building falls $y=16t^2$ in the first $t$ seconds. Find the speed of the object at $t=4$ seconds.
Solution: Try using the points $4$ and $4+h$, where $h$ is an infinitesimally small number.

• when $t=4$, $f(x)=256$
• when $t=4+h$, $f(x)=16(4+h{)}^2$
  Now calculate the slope using these two points, using the limit as h goes to 0: ${\displaystyle \underset{h\to 0}{lim}\frac{16(4+h{)}^2-256}{4+h-4}}$
  Solve this and eventually plug in $h$ (after cancelling the $h$ on the bottom) for a final answer of 128 ft/sec.

Properties of Limits

The Sandwich Theorem