Unit 1: Limits and Continuity

Limits from Graphs & Tables

100

Based on a table where;

f(x) is 5.002 at x=2.999 and 4.998 at x=3.001, 

what is the best approximation for lim{x to 3} f(x)?

5

200

Given a graph of f with a hole at (3, 2) and a solid dot at (3, 4), what is the value of lim{x to 3} f(x)?

2

300

If a table shows f(x) values jumping from 

-625 at x=3.9999 to 5.9999 at x=4.0001, 

what is the right-hand limit lim{x to 4^+}f(x)?

6

400

CHALLENGE: For a piecewise function where the graph of f is a line segment for x < 3 and a parabola for x > 3, if the table shows f(2.999) = 1.601 and f(3.001) = 1.603, what is lim{x to 3} f(x)?

1.6

Analytical Limits (Algebra)

100

Evaluate lim{x to 0} (cos x + 3e^x)/(2e^x) using direct substitution. 

2

200

Simplify lim{x to 9} (x-9)/(sqrt{x}-3)

lim{x to 9} sqrt{x}+3 = 6

300

Simplify f(x) = ((1/x)-1)/(x-1) to find lim{x to 1} f(x)

lim{x to 1} -1/x = -1

400

CHALLENGE: Evaluate the "nested" limit lim{x to 2} f(f(x)) for the piecewise function where f(x)= -x²+3x+3 for x < 2 and f(2) = 6.

-7

Continuity & Discontinuity

100

If lim{x to c} f(x) exists but does not equal f(c), what type of discontinuity is present?

Removable Discontinuity (a hole)

200

For g(x) = (x^2-9)/(4x+12) for x does not equal -3, what value of k makes the function continuous at x = -3?

-3/2

300

A function f is continuous on (-1, 3) but not on [-1, 3]. Which expression could represent f? 

(A) (x+1)/(x-3) or (B) (x+1)(x-3)

(A) (x+1)/(x-3)

400

CHALLENGE: Solve for the constant b that makes f(x) continuous at x=2 if f(x) = e^bx for x less than or equal to 2 and f(x) = 1.5x + b for x > 2.

b is approximately 0.508 and/or b = -1.282

Squeeze Theorem & Theorems

100

 If g(x) less than or equal to f(x) less than or equal to h(x) and both g and h approach 17 as x less than or equal to 5, what is lim{x to 5} f(x)? 

17

200

Which theorem guarantees a solution to f(c) = 0 on [12, 15] if f(12) and f(15) have opposite signs? 

Intermediate Value Theorem (IVT)

300

What must be true about the relationship between f(x), g(x), and h(x) on an interval for the Squeeze Theorem to apply? 

g(x) less than or equal to f(x) less than or equal to h(x) (f must be "trapped" between them)

400

CHALLENGE: If g(x) = (7x-26)/(x-5) and h(x) = (3x+14)/(2x+1) traps f(x), find lim{x to 2} f(x). 

4; Both g(2) and h(2) equal 4

Asymptotes & Infinity

100

If lim{x to5^-} f(x) = infinity, what exists at x=5? 

Vertical Asymptote

200

Find the horizontal asymptote of f(x) = (2x+3)/(x+1). 

y=2

300

For P(t) = (6000)/(40 + 60e^-0.03t), what is the value of P(t) as t to infinity? 

150

400

CHALLENGE: Identify the horizontal asymptote for f(x) = (3x²⁰)(4e^x + 8x²⁰)as x to infinity.

y = 0;  (Because the exponential e^x grows much faster than x²⁰)