Math Problem Statement

The graph of the following function is shown in the figure. Find the largest δ such that if 0 < |x − 2| < δ, then |f(x) − 1| < 0.01 for f(x) = 1 / (x - 1).

Solution

The problem provided is about finding the largest δ\delta such that if 0<x2<δ0 < |x - 2| < \delta, then f(x)1<0.01|f(x) - 1| < 0.01 for the given function f(x)=1x1f(x) = \frac{1}{x - 1}. The graph of the function near x=2x = 2 is shown, and we need to use this to determine the value of δ\delta.

Step-by-Step Solution:

We are dealing with the function: f(x)=1x1f(x) = \frac{1}{x - 1} and we are asked to find the largest δ\delta such that: f(x)1<0.01whenever0<x2<δ.|f(x) - 1| < 0.01 \quad \text{whenever} \quad 0 < |x - 2| < \delta.

  1. Rewriting the inequality: The condition is f(x)1<0.01|f(x) - 1| < 0.01, which means: 1x11<0.01.| \frac{1}{x - 1} - 1 | < 0.01.

    Simplifying the expression: 1x11=1(x1)x1=2xx1.\frac{1}{x - 1} - 1 = \frac{1 - (x - 1)}{x - 1} = \frac{2 - x}{x - 1}. So we now need: 2xx1<0.01.\left| \frac{2 - x}{x - 1} \right| < 0.01.

  2. Solving the inequality: This inequality can be written as: 2xx1<0.01.\frac{|2 - x|}{|x - 1|} < 0.01. This implies that: 2x<0.01x1.|2 - x| < 0.01 |x - 1|.

  3. Graphical Interpretation: We want to find the largest δ\delta such that when xx is close to 2 (but not equal to 2), the function's values satisfy f(x)1<0.01|f(x) - 1| < 0.01.

    The zoomed-in part of the graph near x=2x = 2 shows a range of values where f(x)f(x) is between 0.99 and 1.01. From this visual approximation, it seems that the interval around x=2x = 2 for which this condition holds is small. From the graph, we can estimate that δ0.005\delta \approx 0.005, since this is the range where the function is between 0.99 and 1.01.

Thus, the largest δ\delta is approximately 0.005 based on the graph.


Do you have any questions or need further details?

Here are 5 related questions to deepen understanding:

  1. How can you calculate the exact value of δ\delta algebraically instead of graphically?
  2. What does limx2f(x)=1\lim_{x \to 2} f(x) = 1 imply about the behavior of the function near x=2x = 2?
  3. How does the function behave as xx approaches 1 from the right and left?
  4. What are the implications of finding δ\delta in the context of limits and continuity?
  5. How would the answer change if the inequality was f(x)1<0.001|f(x) - 1| < 0.001?

Tip: Always double-check your results graphically and algebraically to ensure consistency.

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Math Problem Analysis

Mathematical Concepts

Limits
Continuity
Inequalities

Formulas

f(x) = 1 / (x - 1)
|f(x) - 1| < 0.01
|2 - x| / |x - 1| < 0.01

Theorems

Limit definition of continuity
Epsilon-delta definition

Suitable Grade Level

Undergraduate Calculus