Limit Evaluation of f(x) = (1−cos(2x−2))/(2(x−1)^2) as x approaches 1

Math Problem Statement

Use a graph of f to estimate ModifyingBelow lim With x right arrow a f left parenthesis x right parenthesislimx→a f(x) or to show that the limit does not exist. Evaluate f(x) near xequals=a to support your conjecture. Complete parts (a) and (b). f left parenthesis x right parenthesis equals StartFraction 1 minus cosine left parenthesis 2 x minus 2 right parenthesis Over 2 left parenthesis x minus 1 right parenthesis squared EndFractionf(x)= 1−cos(2x−2) 2(x−1)2; aequals=1 Question content area bottom Part 1 a. Use a graphing utility to graph f. Select the correct graph below.. A.

A coordinate system has a horizontal axis from negative 1 to 3 in increments of 0.5 and a vertical axis from 0 to 5 in increments of 1. A periodic curve has amplitude 1 and a period of approximately 3.1. Within one period, the curve reaches a maximum at (0.2, 2.75) and a minimum at (1.8, 0.75). An open circle is plotted at (1, 1.75). All coordinates are approximate. C.

A coordinate system has a horizontal axis from negative 1 to 3 in increments of 0.5 and a vertical axis from 0 to 5 in increments of 1. A periodic curve has amplitude 1 and a period of approximately 3.1. Within one period, the curve reaches a minimum at (1, 0.5) and a maximum at (2.55, 2.5). An open circle is plotted at (1, 0.5). All coordinates are approximate. D.

A coordinate system has a horizontal axis from negative 1 to 3 in increments of 0.5 and a vertical axis from 0 to 5 in increments of 1. An oscillating curve increases in amplitude from left to right, reaching a maximum at (1, 1.0) and then decreases in amplitude. An open circle is plotted at (1, 1.0). All coordinates are approximate. Your answer is correct. Each graph is displayed in a [minus−1,3] by [0,55] window. Part 2 Use the graphing utility to estimate ModifyingBelow lim With x right arrow 1 f left parenthesis x right parenthesislimx→1 f(x). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The limit appears to be approximately 11. (Round to the nearest tenth as needed.) Your answer is correct.B. The limit does not exist. Part 3 b. Evaluate f(x) for values of x near 1 to support your conjecture. 11

x 0.90.9 0.990.99 0.9990.999 1.0011.001 1.011.01 1.11.1 f(x) 0.0003050.000305 0.0003050.000305 0.0003050.000305 0.0003050.000305 0.0003050.000305 0.0003050.000305 (Round to six decimal places as needed.)

Solution

The graph of $f(x) = \frac{1 - \cos(2x - 2)}{2(x - 1)^2}$ shows that as $x$ approaches 1, the function seems to approach a finite value. There is a vertical asymptote at $x = 1$ , indicating some form of discontinuity. The graph does not appear to be oscillating wildly, but the behavior suggests a possible singularity or a limit that could be non-existent at $x = 1$ .

Part 2: Estimating the Limit

Based on the graph, it seems that $\lim_{x \to 1} f(x)$ exists and approaches approximately 1. Therefore, the correct choice in Part 2 is:

A. The limit appears to be approximately $1$ .

Part 3: Evaluating $f(x)$ Near $x = 1$

For values of $x$ near 1, $f(x)$ is evaluated to six decimal places:

$f(0.9) = 0.000305$
$f(0.99) = 0.000305$
$f(0.999) = 0.000305$
$f(1.001) = 0.000305$
$f(1.01) = 0.000305$
$f(1.1) = 0.000305$

This consistent small value near zero supports the conjecture that the limit is approximately 1. Despite the function's form, the actual value for all practical purposes near $x = 1$ stays close to zero, affirming that the limit exists and is around 1.

Additional Questions:

How does the behavior of $\cos(2x - 2)$ affect the shape of the graph?
Why does the denominator $(x - 1)^2$ cause a vertical asymptote at $x = 1$ ?
How could you modify the function to avoid the singularity at $x = 1$ ?
What does the small variation in $f(x)$ near $x = 1$ suggest about the function's continuity?
How would $f(x)$ behave for values of $x$ much further away from 1?

Tip:

When evaluating limits, check for discontinuities or undefined points by factoring or simplifying the function around the limit point.

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Math Problem Analysis

Mathematical Concepts

Limits
Trigonometric Functions
Asymptotic Behavior

Formulas

f(x) = (1−cos(2x−2))/(2(x−1)^2)
limx→1 f(x)

Theorems

Limit Definition
L'Hopital's Rule

Suitable Grade Level

Undergraduate (Calculus I/II)

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