U1: Limits and Continuity

Limits

What is a limit and how to find it

Limit: If f(x) becomes close to a unique number L as x approaches c from either side, then the limit of f(x) as x approaches c is L.

● A limit refers to the y-value of a function

● The general limit exists when the right and left limits are the same/equal each other.

● DNE = does not exist.

Examples of estimating a limit numerically

● Example 2: Given lim (3x - 2), find what L would be when you plug in the constant of 2.

Lim f(x) = L

x → c

Lim

f(3(2) - 2) = 4

x → 2

*2 is the limit.

When limits don’t exist

When the Left limit ≠ Right limit, then the limit is said to not exist.

● In the picture below, you can tell that the two limits don’t equal each other, thus the answer to this limit is DNE.

Unbounded Behavior:

lim = 1/x2

x→0

Evaluating Limits Analytically

Limits Theorem

Limits at Infinity

● If m<n, then the limit equals 0

● If m=n, then the limit equals a/b

● If m>n, then the limit DNE

Lim = (axm)/(bxm)

x→ ±∞

Finding Vertical Asymptotes

The only step you have to do is set the denominator equal to zero and solve.

● Example:

f(x) = (x - 2)/(x2 - 4) = (x - 2)/((x + 2)(x - 2))

○ (x+2)(x-2) = 0 → x = 2, -2

■ 2 is a removable hole while -2 is the non-removable vertical asymptote.

Finding Horizontal Asymptotes

Use the two terms of the highest degree in the numerator and denominator

lim = (x - 2)/(x2 - 4)

x → ∞

● Example:

○ x and x2 are the two terms of the highest degree in the numerator and denominator respectively. After finding it, use the limits at infinity rule to determine the limit.

Intermediate Value Theorem

A continuous function on a closed interval cannot skip values.

● f(x) must be continuous on the given interval [a,b]

● f(a) and f(b) cannot equal each other.

● f(c) must be in between f(a) and f(b)

Example #1: Apply the IVT, if possible on [0, 5] so that f(c) = 1 for the function f(x) = x2 + x + 1

1) f(x) is continuous because it is a polynomial function.

2) f(a) = f(0) = 1

f(b) = f(5) = 29

3) By the IVT, there exists a value c where f(c)=1 since 1 is between -1 and 29.

Example #2:

1) For 0 < t < 60, must there be a time t when v(t) = -5?

2) f(a) = f(0) = -20

f(b) = f(60) =10

3) By the IVT, there is a time t where v(t) = -5 on the interval [0, 60] since -20 < -5 < 10

The Squeeze Theorem

h(x) ≤ f(x) ≤ g(x)

lim h(x) = lim g(x) = L

x→a x→a

therefore,

lim h(x) = L

X→a

that means f(x) equals h(x) and g(x)