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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 2 - Limits and ContinuityProblem 2.3.32
Chapter 2, Problem 2.3.32

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

Verified step by step guidance
1
Step 1: Begin by finding the limit L = lim x→c f(x). Substitute c = -1 into the function f(x) = -3x - 2 to find L. This involves calculating f(-1).
Step 2: Calculate f(-1) by substituting x = -1 into the function: f(-1) = -3(-1) - 2. Simplify this expression to find the value of L.
Step 3: With L determined, we need to find δ > 0 such that |f(x) - L| < ε whenever 0 < |x - c| < δ. Start by expressing |f(x) - L| in terms of x.
Step 4: Substitute f(x) = -3x - 2 and L into the inequality |f(x) - L| < ε. This gives |(-3x - 2) - L| < 0.03. Simplify this expression to find a condition on x.
Step 5: Solve the inequality |(-3x - 2) - L| < 0.03 for x, which will involve isolating |x + 1|. Determine δ such that 0 < |x + 1| < δ satisfies the inequality.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limit of a Function

The limit of a function describes the value that the function approaches as the input approaches a certain point. In this case, we need to find L = lim x→c f(x), which means evaluating the function f(x) at the point c and determining the value it approaches as x gets closer to c.
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Epsilon-Delta Definition of a Limit

The epsilon-delta definition formalizes the concept of limits. It states that for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε. This means we can make the function's output as close to L as desired by restricting x sufficiently close to c.
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Finding δ for Given ε

To find δ for a given ε, we analyze the inequality |f(x) - L| < ε. This involves manipulating the expression to isolate |x - c|, allowing us to determine a suitable δ that ensures the function's output remains within the ε range of L when x is within δ of c.
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Related Practice
Textbook Question

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


llimx→0 (x −x cos x) / sin² 3x

Textbook Question

In Exercises 1–4, say whether the function graphed is continuous on [−1, 3]. If not, where does it fail to be continuous and why?

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Textbook Question

Horizontal and Vertical Asymptotes

Assume that constants a and b are positive. Find equations for all horizontal and vertical asymptotes for the graph of y = (√ax² + 4) / (x―b) .

Textbook Question

Finding Limits of Differences When x → ±∞


Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)


lim x → −∞ (2x + √(4x² + 3x − 2))

Textbook Question

Slope of a Curve at a Point


In Exercises 7–18, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P.


y=7−x², P(2,3)

Textbook Question

Limits as x → ∞ or x → −∞


The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.


lim x→∞ √(x² + 1) / (x + 1)