Note an important special case of the Intermediate Value Theorem:
If f is continuous on the closed interval [a,b], and f (a) and f (b) have opposite signs, then f has a
zero in that interval (there is a value, c, in [a,b] where f (c) = 0).
(3) The Continuous Functions Theorem. If functions f and g are both continuous at x = c, then so are
the following functions:
(a) kf, where k is a constant;
(b) f ± g;
(c) f · g;
(d) provided that g(c) ≠ 0.
EXAMPLE 32
Show that has a root between x = 2 and x = 3.SOLUTION: The rational function f is discontinuous only at and f (3) = 1.
Since f is continuous on the interval [2,3] and f (2) and f (3) have opposite signs, there is a value,
c, in the interval where f (c) = 0, by the Intermediate Value Theorem.Chapter Summary
In this chapter, we have reviewed the concept of a limit. We’ve practiced finding limits using
algebraic expressions, graphs, and the Squeeze (Sandwich) Theorem. We have used limits to find
horizontal and vertical asymptotes and to assess the continuity of a function. We have reviewed
removable, jump, and infinite discontinuities. We have also looked at the very important Extreme
Value Theorem and Intermediate Value Theorem.
Practice Exercises
Part A. Directions: Answer these questions without using your calculator.
1.
(A) 1
(B) 0
(C)
(D) −1
(E) ∞