Unit 2 – Limits

2.1 Tangent and Velocity problemsThe word tangent is derived from the Latin word tangens, which means _________________. Thus a tangent to a curve is a line that _______________________________. In other words, a tangent line should have the same direction as the curve at the point of contact. How do we make sure that it is precise?

Ex1) Find an equation of the tangent line to the parabola y=x2 at the point P(1,1).

mPQ=x2−1x−1

x mPQ x mPQ

2 0

1.5 0.5

1.1 0.91.01 0.991.001 0.99

Velocity Problemex) Suppose that a ball is dropped from the upper observation deck of the CN Tower in Toronto, 450m above the ground. Find the velocity of the ball after 5 seconds.

By Galileo, the position of any freely falling object:s(t)= 4.9t2

Since velocity=change∈positiontimeelapsed ,

v (5 )=4.9 (5.001 )2−4.9 (5 )2

5.001−5=49.0049

v(5)=49m/s2.2 The limit of a function

limx→2

(x2−x+2 )=4

It is also read as “f(x) approaches L as x approaches a”

Ex1) Guess the value of limx→1

x−1x2−1

x<1 f (x) x>1 f (x)0.5 1.50.9 1.10.99 1.010.999 1.0010.9999 1.0001

Ex2) limt →0

√ t 2+9−3t2

t √t 2+9−3t 2

±1.0

±0.5

±0.1

±0.05

±0.01

If you make t sufficiently small, you will get 0 on your calculator. It does NOT mean answer is really 0. The calculator gave false values because √ t2+9 becomes really close to 3, and your calculator does not carry enough digits.

One-Sided Limits

Ex1)

Ex2)

Vertical Asymptote: at least one of following six statements should be true:

Ex)

2.3 Calculating limits using the limit laws

Example)

Ex)

Ex)

Ex)

Ex)

Ex)

Ex)

Ex)

2.5 Continuity

Note that definition 1 requires three conditions:

1.

2.

3.

Ex) At which numbers is f discontinuous? Why?

Ex) Where are each of the following functions discontinuous?

Removable discontinuity Infinite discontinuity Jump discontinuity

Ex)

Ex)

Ex)

2.6 Limits at infinity; Horizontal asymptotes

Investigate function f ( x )= x2−1x2+1

Then, we write as:

Ex) Find infinite limits, limits at infinity, and asymptotes for the function f.

Ex) Evaluate

2.7 Derivatives and rates of change

In section 2.1, we found instantaneous velocity and slope of tangent line by

m=f ( x+∆ x )−f (x )

∆ x

The slope gets more precise as __________________________

Ex) Find the equation of the tangent line to f ( x )=x2 at (-2,4)

Ex) Find the derivative of the function y=x2−8 x+9 at the number a.

2.8 Derivative as a function

f ' (x) is called the derivative of function f(x), and also written as ___________________________

Ex)

Failing to be differentiable

Higher orders of derivatives: