Finding Deltas Algebraically


Each of Exercises 15–30 gives a function f(x) and numbers L, c, and ε>0. In each case, find the largest open interval about c on which the inequality |f(x)−L| <ε holds. Then give a value for δ>0 such that for all x satisfying 0 < |x−c| < δ, the inequality |f(x)−L| < ε holds.


f(x) = 1/x, L = 1/4, c = 4, ε = 0.05

Theory and Examples

If x⁴ ≤ f(x) ≤ x² for x in [−1,1] and x² ≤ f(x) ≤ x⁴ for x < - 1 and x > 1, at what points c do you automatically know limx→c f(x)? What can you say about the value of the limit at these points?

Theory and Examples

Suppose that g(x) ≤ f(x) ≤ h(x) for all x≠2 and suppose that lim x→2 g(x) = lim x→2 h(x) = −5. Can we conclude anything about the values of f, g, and h at x = 2? Could f(2) = 0? Could limx→2 f(x)=0? Give reasons for your answers.

Centering Intervals About a Point

In Exercises 1–6, sketch the interval (a,b), on the x-axis with the point c inside. Then find a value of δ>0 such that a < x < b whenever 0 < |x−c| < δ.

a=4/9, b=4/7, c=1/2

Finding Deltas Graphically

In Exercises 7–14, use the graphs to find a δ>0 such that |f(x)−L| <ε whenever 0< |x−c| <δ.

Finding Deltas Algebraically

Each of Exercises 15–30 gives a function f(x) and numbers L, c, and ε>0. In each case, find the largest open interval about c on which the inequality |f(x)−L| <ε holds. Then give a value for δ>0 such that for all x satisfying 0 < |x−c| < δ, the inequality |f(x)−L| < ε holds.

f(x) = mx, m > 0, L = 2m, c = 2, ε = 0.03

Using the Formal Definition

Each of Exercises 31–36 gives a function f(x), a point c, and a positive number ε. Find L = lim x→c f(x). Then find a number δ > 0 such that |f(x)−L| < ε whenever 0 < |x−c| < δ.

f(x) = −3x − 2, c = −1, ε = 0.03

[Technology Exercise] Grinding engine cylinders Before contracting to grind engine cylinders to a cross-sectional area of 9in², you need to know how much deviation from the ideal cylinder diameter of c = 3.385in. you can allow and still have the area come within 0.01in² of the required 9in². To find out, you let A=π(x/2)² and look for the largest interval in which you must hold x to make |A − 9| ≤ 0.01. What interval do you find?

Finding Limits Graphically

Which of the following statements about the function y = f(x) graphed here are true, and which are false?

a. limx→−1+ f(x) = 1