Use the graph of ff in the figure to determine the values of xx in the interval (−3,5)\left(-3,5\right) at which f fails to be continuous. Justify your answers using the continuity checklist.

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

Problem 2.R.5

Chapter 2, Problem 2.R.5

Use the graph of $f$ in the figure to determine the values of $x$ in the interval $open paren negative 3 comma 5 close paren$ at which f fails to be continuous. Justify your answers using the continuity checklist.
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Verified step by step guidance

Step 1: Recall the definition of continuity. A function f(x) is continuous at a point x = c if the following three conditions are met: (1) f(c) is defined, (2) $ \lim_{{x \to c^-}} f(x) = \lim_{{x \to c^+}} f(x) $, and (3) $ \lim_{{x \to c}} f(x) = f(c) $.

Step 2: Examine the graph of f(x) over the interval $(-3, 5)$ to identify any points where the function might be discontinuous. Look for breaks, jumps, or holes in the graph.

Step 3: For each potential point of discontinuity, check if f(c) is defined. If f(c) is not defined, then f is not continuous at that point.

Step 4: For each potential point of discontinuity where f(c) is defined, calculate the left-hand limit $ \lim_{{x \to c^-}} f(x) $ and the right-hand limit $ \lim_{{x \to c^+}} f(x) $. If these limits are not equal, f is not continuous at that point.

Step 5: If both the left-hand and right-hand limits exist and are equal, check if they equal f(c). If they do not, f is not continuous at that point.

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Key Concepts

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

Continuity of Functions

A function is continuous at a point if three conditions are met: the function is defined at that point, the limit of the function as it approaches that point exists, and the limit equals the function's value at that point. Understanding these conditions is crucial for identifying points of discontinuity in a function's graph.

Types of Discontinuities

Discontinuities can be classified into three main types: removable, jump, and infinite. A removable discontinuity occurs when a function can be made continuous by redefining a point, a jump discontinuity involves a sudden change in function value, and an infinite discontinuity occurs when the function approaches infinity at a point. Recognizing these types helps in analyzing the graph effectively.

The Continuity Checklist

The continuity checklist is a systematic approach to determine if a function is continuous at a given point. It involves checking if the function is defined at that point, evaluating the limit from both sides, and ensuring that the limit matches the function's value. This checklist is essential for justifying the identification of discontinuities in the given interval.

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Related Practice

Textbook Question

The height above the ground of a stone thrown upwards is given by s(t), where t is measured in seconds. After 1 second, the height of the stone is 48 feet above the ground, and after 1.5 seconds, the height of the stone is 60 feet above the ground. Evaluate s(1) and s(1.5), and then find the average velocity of the stone over the time interval [1, 1.5].

Textbook Question

Determine the following limits.

lim x→∞ (5 + (cos⁴ x) / (x² + x + 1))

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Textbook Question

Evaluate ${$\displaystyle\[\lim$_{x$\to\]\infty$}{f(x)}}$ and ${$\displaystyle$$\lim$_{x$\to$-$\infty$}{f(x)}}$ .