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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 2.2.21
Chapter 2, Problem 2.2.21

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 if x<4
3 if x=4; a=4
x+1 if x>4

Verified step by step guidance
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Step 1: Understand the piecewise function f(x). It is defined as f(x) = \(\sqrt{x}\) for x < 4, f(x) = 3 for x = 4, and f(x) = x + 1 for x > 4.
Step 2: Sketch the graph of each piece of the function. For x < 4, plot the curve of \(\sqrt{x}\), which is a part of the square root function starting from the origin and increasing as x approaches 4.
Step 3: At x = 4, plot the point (4, 3) since f(x) = 3 when x = 4. This is a single point on the graph.
Step 4: For x > 4, plot the line x + 1, which starts at (4, 5) and increases linearly. Note that this line does not include the point (4, 5) since x > 4.
Step 5: Analyze the graph to make conjectures about the limits. Observe the behavior of the function as x approaches 4 from the left and right, and compare these to the value of f(4).

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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 three distinct definitions depending on whether x is less than, equal to, or greater than 4. Understanding how to evaluate and graph these segments is crucial for analyzing the function's behavior at specific points.
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Piecewise Functions

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 at a point. If both limits exist and are equal, the overall limit (lim x→a f(x)) exists and equals that common value.
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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 f(x) at x=4, we need to check if lim x→4^−f(x) equals lim x→4^+f(x) and if both equal f(4). If they do not match, the function is discontinuous at that point, which affects the conjecture about the function's values.
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Related Practice
Textbook Question

Suppose f(x) lies in the interval (2, 6). What is the smallest value of ε such that |f (x)−4|<ε?

Textbook Question

Determine limx→∞f(x)\(\lim\)_{x\(\rightarrow\]\infty\)}f\(\left\)(x\(\right\)) and limx→−∞f(x)\(\lim\)_{x\(\rightarrow\)-\(\infty\)}f\(\left\)(x\(\right\)) for the following functions. Then give the horizontal asymptotes of ff (if any).


f(x)=4x20x+1f\(\left\)(x\(\right\))=\(\frac{4x}{20x+1}\)

Textbook Question

Determine limx→∞f(x)\(\lim\)_{x\(\rightarrow\]\infty\)}f\(\left\)(x\(\right\)) and limx→−∞f(x)\(\lim\)_{x\(\rightarrow\)-\(\infty\)}f\(\left\)(x\(\right\)) for the following functions. Then give the horizontal asymptotes of ff (if any).


f(x)=3x3−7x4+5x2f\(\left\)(x\(\right\))=\(\frac{3x^3-7}{x^4+5x^2}\)

Textbook Question

Determine the interval(s) on which the following functions are continuous. 

p(x)=4x^5−3x^2+1

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Textbook Question

Suppose |f(x) − 5|<0.1 whenever 0<x<5. Find all values of δ>0 such that |f(x) − 5|<0.1 whenever 0<|x−2|<δ.

Textbook Question

Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right? 


f(z)=(z−1)^3/4