Limits and Continuity

Thu Mai, Michelle Wong, Tam Vu

What are Limits?

Limits are built upon the concept of infinitesimal. Instead of evaluating a function at a certain x-value, limits ask the question, “What value does a function approaches as its input and a constant becomes infinitesimally small?” Notice how this question does not depend upon what f(c) actually is. The notations for writing a limit as x approaches a constant of the function f(x) is:

Where c is the constant and L (if it is defined) is the value that the function approaches.

Evaluating Limits: Direct Substitution

Sometimes, the limit as x approaches c of f(x) is equal to f(c). If this is the case, just directly substitute in c for x in the limit expression, as shown below.

Dividing Out Technique1. Always start by seeing if the substitution method works.2. If, when you do so, the new expression obtained is an indeterminate form such as

0/0… try the dividing out technique!3. Because both the numerator an denominator are 0, you know they share a similar

factor.4. Factor whatever you can in the given function.5. If there is a matching factor in the numerator and denominator, you can cross thru

them since they “one out.”6. With your new, simplified function attempt the substitution method again. Plug

whatever value x is approaching in for x.7. The answer you arrive at is the limit.

*Note: You may need to algebraically manipulate the function.

RationalizingSometimes, you will come across limits with radicals in fractions.

Steps1. Use direct substitution by plugging in zero for x.2. If you arrive at an undefined answer (0 in the denominator) see if there are any

obvious factors you could divide out.3. If there are none, you can try to rationalize either the numerator or the

denominator by multiplying the expression with a special form of 1.4. Simplify the expression. Then evaluate the rewritten limit.

Ex:

Squeeze Theorem

The Squeeze Theorem states that if h(x) f(x) g(x), and

then

Special Trig Limits(memorize these)

h is angle in radiansarea of blue: cos(h)sin(h)/2

area of pink: h/2area of yellow: tan(h)/2

Since

by the Squeeze Theorem we can say that

Special Trig Limits Continued

Continuity and DiscontinuityA function is continuous in the interval [a,b] if there does not exist a c in the interval [a,b] such that:1) f(c) is undefined, or2) , or 3)

The following functions are discontinuous b/c they do not fulfill ALL the properties of continuity as defined above.

Removable vs Non-removable Discontinuities • A removable discontinuity exists at c if f can be made continuous by redefining f(c). • If there is a removable discontinuity at c, the limit as xc exists; likewise if there is a non-removable discontinuity at c, the limit as xc does not exist.

For this function, there is a removable discontinuity at x=3; f(3) = 4 can simply be redefined as f(3) = 2 to make the function continuous. The limit as x3 exists.

For this function, there is a non-removable discontinuity at x=3; even if f(3) is redefined, the function will never be continuous. The limit as x3 does not exist.

Intermediate Value Theorem

The Intermediate Value Theorem states that if f(x) is continuous in the closed interval [a,b] and f(a) M f(b), then at least one c exists in the interval [a,b] such that:

f(c) = M

When do limits not exist?

then…

If

Vertical Asymptotesf(x) and g(x) are continuous on an open interval containing c. if f(c) is not equal to 0 and g(c)= 0 and there’s an open interval with c which g(x) is not 0 for all values of x that are not c, then…..

There is an asymptote at x = c for

Properties of Limits

Sum or Difference

Scalar Multiple

Product

Quotient

Power

Let b and c be real numbers, n be a positive integer, f and g be functions with the following limits.

Limits Substitution

With limits substitution (informally named so by yours truly), if

then

This is useful for evaluating limits such as:

How Do Limits Relate to Derivatives?

What is a derivative?• The derivative of a function is defined as that function’s INSTANT rate of change.

Applying Prior Knowledge:• As learned in pre-algebra, the rate of change of a function is defined by:

Apply Knowledge of Limits:• Consider that a limit describes the behavior of a function as x gets closer and closer to a point on a function from both left and right.• describes a function’s rate of change. To find the function’s INSTANT rate of

change, we can use limits.• We can take:

WHY? As the change in x gets closer and closer to 0, we can more accurately predict the function’s INSTANT rate of change, and thus the function’s derivative.

Δy Δx

Δy Δx

Δy Δx

Δx 0 lim

How Do Limits Relate to Derivatives?

• Δy Δx

• Consider that can be rewritten as .

• y2 – y1 Δx

(x, f(x))

(x+ Δx, f(x+ Δx))Analyze the graph. Notice that the change in y between any two points on a function is f(x+ Δx) – f(x). Thus:

Δy Δx

y2 – y1 Δx

f(x+ Δx) – f(x)Δx

==

Δy Δx

Δx 0 limSo can be rewritten as

.

f(x+ Δx) – f(x)Δx

limΔx0

Therefore, the derivative of f(x) at x is given by:

f(x+ Δx) – f(x)Δx

limΔx 0