Calculus Quiz 1: Limits and Continuity

How to calculate IROC?

Use progressive smaller intervals and round; make sure to find from left and right if general limit.

How to calculate AROC?

Slope

3 situations in which limits DO NOT exist

1. Limits different L and R

2. Oscillating behavior

3. Unbounded behavior* can be defined, but not a FORMAL limit

The 7 properties of limits

• Constant rule

• Identity rule (x=y)

• Coefficient rule

• Sum/difference rule

• product/quotient rule

• power/root rule

• composition rule

Composition: Internal limit DNE

Find both L and R values; denote if from above (+) or below (-). Then plug these in for limits of the other function; if equal, defined.

Composition: External limit DNE

Find the L and R limit of the first function; if both approach from above/below, then its equal to the one directional limit. Otherwise DNE.

Finding Limits Analytically

Plug in; if that isn’t going to work then factor and cancel and then.

• Sinx/x for lim→0 is 1

• 1-cosx/x for lim→0 is 0

When a limit using the l or r of value has asymptote at value

Analyze the values from that side; then decide if inf or -inf

What to do for complex fractions

multiply by LCM (denominators multiplied)

when you have a root expression in a fraction

multiply by the conjugate to rationalize and then solve

When you have an absolute value function

Find the l and r, then decide what you need to use based on the intervals

A root of a function to a power

turns into the absolue value of that function

if you get smth/0 or inf/inf

use the left and right limits; if they arent equal DNE

horizontal asymptotes (lim) as x=inf

• If bottom heavy, 0

• If same, leading coefficient ratio

• If top heavy, look at leading terms; behaves like quotient of those terms

Intermediate Value Theorem

• For a continuous function, f(a) and f(b) will run through every value between them, and there is a input to get a specific one.