Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 2 - Limits

Problem 2.2.19

Chapter 2, Problem 2.2.19

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = {x^2+1 if x≤−1
3 if x>−1; a=−1

Verified step by step guidance

Step 1: Identify the piecewise function. The function f(x) is defined as f(x) = x^2 + 1 for x ≤ -1 and f(x) = 3 for x > -1. This means the function has two different expressions depending on the value of x relative to -1.

Step 2: Sketch the graph of f(x). For x ≤ -1, plot the graph of y = x^2 + 1, which is a parabola opening upwards with its vertex at (0, 1). For x > -1, plot the horizontal line y = 3. Note the point of transition at x = -1.

Step 3: Determine f(a). Since a = -1, use the expression for x ≤ -1, which is f(x) = x^2 + 1. Therefore, f(-1) = (-1)^2 + 1 = 2.

Step 4: Evaluate the left-hand limit, \( \lim_{x \to -1^-} f(x) \). As x approaches -1 from the left, f(x) follows the expression x^2 + 1. Thus, \( \lim_{x \to -1^-} f(x) = (-1)^2 + 1 = 2 \).

Step 5: Evaluate the right-hand limit, \( \lim_{x \to -1^+} f(x) \). As x approaches -1 from the right, f(x) is constant at 3. Thus, \( \lim_{x \to -1^+} f(x) = 3 \).

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Key Concepts

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

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, f(x) has two distinct definitions: one for x ≤ -1 and another for x > -1. Understanding how to evaluate piecewise functions at specific points is crucial for analyzing their behavior and limits.

Limits

Limits describe the behavior of a function as the input approaches a certain value. The left-hand limit (lim x→a^−f(x)) and right-hand limit (lim x→a^+f(x)) are essential for determining continuity and the overall limit (lim x→a f(x)). Evaluating these limits helps in understanding how the function behaves around the point of interest.

Continuity

A function is continuous at a point if the limit as x approaches that point equals the function's value at that point. For the function f(x) at a = -1, checking continuity involves comparing f(-1) with the left-hand and right-hand limits. If these values do not match, the function is discontinuous at that point.

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Related Practice

Textbook Question

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→0 x^2=0 (Hint: Use the identity √x2=|x|.)

Textbook Question

Suppose f(x)→100 and g(x)→0, with g(x)<0 as x→2. Determine lim x→2 f(x) / g(x).

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Textbook Question

Determine the following limits.

lim x→∞ sin x / e^x

Textbook Question

Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.

f(x) = x^2−25 / x−5; a=5

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Textbook Question

Determine the following limits at infinity.

lim x→∞ (5 + 1/x +10/x^2)

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Textbook Question

Let f(x) =x^2−2x+3.

a. For ε=0.25, find the largest value of δ>0 satisfying the statement

|f(x)−2|<ε whenever 0<|x−1|<δ.