AP Calc Unit 1: Limits & Continuity

What is the first thing you should do when finding a limit?

Use direct substitution

• If you get a finite answer, then that is your limit.

What should you do if you use direct substitution and get a #/0?

• This means you have a VA @ that x-value

• Find the left-hand limit using numbers just to the left (_.999)

• Find right-hand limit using just numbers to the right in numbers just to the right (_.01)

• Compare your limits, if they are the same then you have a limit!!

What should you do if you get direct substitution and get 0/0 or ∞/∞?

• Use algebra methods or L’Hospital

1. Factor, simplify, plug

2. Divide (Long or Synthetic)

3. Multiply by conjugate (Radicals)

4. Multiply by LCD

Squeeze Theorem

If h(x)≤f(x)≤g(x) for all x in an open interval containing c and the limit of h(x) and g(x) are equal, then f(x) also has the same limit.

How do you find limits at infinity?

• Also shows end behavior/VA

• Degree of Num<Degree of Denom: f(x)=0

• Degree of Num=Degree of Denom:

  f(x)=ratio b/t leading coefficients

• Degree of Denom>Degree of Num:

  f(x)= ± ∞

How do we know if f(x) is continuous?

Intermediate Value Theorem (IVT)

There is such value c such that f(c)= _, and f(a)≤f(c)≤f(b).

Ranking of Dominance

1. Exponential (Larger Base)

2. Power/Polynomial/Root (Larger Exponent)

3. Logs (Larger Base)

4. Sinx/Cosx