Painless Calculus

Christina Pawlowski-Polanish

• Publisher: Barrons Educational Services
• Year: 2021
• ISBN:
  9781506273204
• Topic: Mathematics
• Subtopic: Calculus
• Index: Mathematics

Learning at home is now the new normal. Need a quick and painless refresher? Barron's Painless books make learning easier while you balance home and school. Teaches basic algebra, exponents and roots, equations and inequalities, and polynomials.Titles in Barron's extensive Painless Series cover a wide range of subjects, as they are taught at middle school and high school levels. Perfect for supporting Common Core Standards, these books are written for students who find the subjects somewhat confusing, or just need a little extra help. Most of these books take a lighthearted, humorous approach to their subjects, and offer fun exercises including puzzles, games, and challenging "Brain Tickler" problems to solve. Bonus Online Component: includes additional games to challenge students, including Beat the Clock, a line match game, and a word scramble.

Chapter 1

Limits and Continuity

The concept of a limit plays a vital role in calculus. A limit will help to evaluate expressions that normally would be undefined or indeterminate.

MATH TALK!

An expression can be undefined in a few ways, such as a 0 in the denominator or approaching ±∞. Indeterminate is different from undefined. Indeterminate means the expression may still have a value, but alternative methods of solution need to be implemented. Indeterminate forms may be

, and others.

Definition of a Limit

Given a function f (x), the limit of f (x) as x approaches c is a real number L if f (x) can be made arbitrarily close to L by having x approach close to c (but not equal to c).

This is notated in the table below.

Notation	Read as
	The limit as x approaches c of f (x) equals L

If the limit equals a value L, the limit exists. It is understood that the values of f (x) approach L as x approaches c from the left and from the right.

This is notated in the table below.

Notation	Read as
	The limit as x approaches c from the left of f (x) equals L
	The limit as x approaches c from the right of f (x) equals L

MATH TALK!

Evaluating a limit from the left means that the x-values are arbitrarily close to c but less than c. Evaluating a limit from the right means that the x-values are arbitrarily close to c but greater than c.

For example, consider the table of values below for f (x) = x².

The

because starting on the left of the table and as x moves closer to 3, f (x) ...