Calculus I Assignment # 1

This assignment covers

A: Algebraic and Trigonometric Limits

B: Limits on Restricted Domains

C: Continuity

QUESTIONS

A) Evaluate These Limits

If the limit does not exist, state why.

 1) 2) 3) 4) 5) 6)

.

B) Limits on Restricted Domains (piecewise defined functions)

Discuss restrictions on the domains and the limits for these functions.

 1) 2) 3)

.

C) Continuity

1) Discuss the discontinuities of these functions:
For removeable jumps, find the limit.

 1) 2) 3) 4)

.

2) Find values for c and/or c and d so that f(x) is continuous for all Real numbers.

 1) 2) 3) 4)

.

SOLUTIONS

A) Evaluate These Limits

If the limit does not exist, state why.

 1) 2) 3) 4) 5) 6)

.

B) Limits on Restricted Domains (piecewise defined functions)

Discuss restrictions on the domains and the limits for these functions.

 1) so limit = 7 2) so limit = 1 3) therefore, limit doesn't exist.

.

C) Continuity

1) Discuss the discontinuities of these functions:
For removeable jumps, find the limit.

 1) is restricted to -3 [ x [ 3 and x 4 - 16 has vertical asymptotes at x = ± 2, so the function is continuous on: -3 [ x < -2 and -2 < x < 2 and 2 < x [ 3. 2) is continuous for x m 0 and x² - 1 has vertical asymptotes at x = ± 1 the function is continuous on x m 0, x ! 1. 3) continuous for all Real numbers, x ! 2 removeable jump at x = 2, y = 1/12. 4) -2 [ x [ 3 ensures that the expression is always positive or zero so that the square root exists.

.

2) Find values for c and/or c and d so that f(x) is continuous for all Real numbers.

 1) set them equal, solve for c = 5/2 or 2.5 2) set them equal, solve for c = - 2. 3) Now we solve the system of 2 equations for c = -2 and d = -6. 4) (c - 2)(c - 1) = 0 gives solution c = 1 or 2.

.

.

.

(all content of the MathRoom Lessons © Tammy the Tutor; 2002 - 2005).