SINGLE VARIABLE CALCULUS. PART I: DIFFERENTIATION. PART A: DEFINITION AND BASIC RULES. UNIT 0: LIMITS REVIEW. PART 1: Introduction to limits

SINGLE VARIABLE CALCULUS

PART I: DIFFERENTIATION

PART A: DEFINITION AND BASIC RULES
PART 1: INTRODUCTION TO LIMITS
UNIT 0: LIMITS REVIEW

I. Definition of left-hand and right-hand side limits

                           
Suppose $f(x)$ gets really close to $R$ for values of $x$ that get really close to (but are not equal to) a from the right. Then we say $R$ is the right-hand limit of the function $f(x)$ as x approaches a from the right.

        We write:

         or

Similarly, with $L$ on the left-hand side, we have the left-hand side limit.

II. Possible limit behaviors

    There are many possible limits to behaviors.

        · The right-hand and left-hand limits may both exist and be equal.

        · The right-hand and left-hand limits may both exist but may fail to be equal.

        · A right- and/or left-hand limit could fail to exist due to blowing up to ±∞. (Example: Consider the function $$1/x$$ near $x=0$.) In this case, we either say the limit blows up to infinity. We also say that the limit does not exist because ∞ is not a real number!

        · A right- and/or left-hand limit could fail to exist because it oscillates between many values and never settles down. In this case, we say the limit does not exist.

III. Definition of the Limit

1. The Limit in Words

If a function $f(x)$ approaches some value $L$ as $x$ approaches a from both the right and the left, then the limit of $f$(x) exists and equals $L$.

2. The Limit in Symbols

    If:

    then:

    Alternatively,

    Remember that $x$ is approaching $a$ but does not equal $a$.

IV. The Limit Laws:

    Suppose

    Then we get the following limit laws:

    Limit Law for Addition: 

    Limit Law for Subtraction: 

    Limit Law for Multiplication: 

    We also have a part of the Limit Law for Division: