Dr. Zs Math151 Handout #2.4 [The Precise Denition of a Limit]
By Doron Zeilberger
Problem Type 2.4.1 :
Use the given graph of f (x) to nd a number such that |f (x) f (a)| <
whenever |x a| < .
(A graph is given with dashed lines at x = a and y = f (a), and
Dr. Zs Math151 Handout #2.1 [The Tangent and Velocity Problems]
By Doron Zeilberger
Problem Type 2.1.1 : The point P (a, f (a) lies on the curve y = f (x). (a) If Q is the point
(x, f (x), use your calculator to nd the slope of the secant line P Q (correc
Dr. Zs Math151 Handout # 2.2 [The Limit of a Function] By Doron Zeilberger Problem Type 2.2.1 : Given a graph of a function, nd the limits from the left, limit from the right, limit (if it exists), at various points, as well as some function values. Examp
Dr. Zs Math151 Handout #2.5 [Continuity] By Doron Zeilberger Problem Type 2.5.1 : Explain why the function is discontinuous at x = a: f (x) = Expression(x), if x = a; Number, if x = a.
Example Problem 2.5.1: Explain why the function is discontinuous a the
Dr. Zs Math151 Handout # 2.3 [Calculating Limits Using the Limit Laws]
By Doron Zeilberger
Problem Type 2.3.1: Evaluate the limit if it exists:
lim f (x)
xa
Example Problem 2.3.1 : Evaluate the limit if it exists:
x2 + 5x + 4
x4 x2 + 3x 4
lim
Steps
Exampl
Dr. Zs Math151 Handout #2.6 [Limits at Innity: Horizontal Asymptotes] By Doron Zeilberger Problem Type 2.6.1: Find T OP (x) x BOT T OM (x) lim where both T OP and BOT T OM are polynomials in x. Example Problem 2.6.1: x3 + 5x x 2x3 x2 + 4 lim , ,
Steps 1.
Dr. Zs Math151 Handout #2.8 [Derivatives]
By Doron Zeilberger
Problem Type 2.8.1 : Find f (a) if f (variable) = Expression(variable).
Example Problem 2.8.1: Find f (a) if
f (s) =
s+3
2s + 1
Steps
Example
1. This is exactly like Problem 4.1 (section 2.7),
Dr. Zs Math151 Handout #2.7 [Tangents, Velocities, and Other Rates of Change] By Doron Zeilberger Problem Type 2.7.1 : Find an equation for the tangent line to the curve at the given point. y = f (a), (a, f (a). Example Problem 2.7.1: Find an equation for