Preparing for a Semester Test: Find the Answers!

[badatmaths]

EDIT: I actually solved the first problem correctly...I must've entered it in wrong earlier. I could still use help with the second problem though.

I have my first test of the semester tomorrow and I'm just trying to get everything I don't fully understand out of the way tonight. Any help is appreciated :)

Homework Statement

[STRIKE]1. For the limit below, find values of δ that correspond to the ε values. ε = 1[/STRIKE]

2. Use the given graph of f(x) = 1/x to find a number δ that fulfills the following condition.

Homework Equations

1.

2.

The Attempt at a Solution

[STRIKE]1. lim x->1 = 0
Since ε = 1, y=(0+1) and (0-1)

y = 1 and -1
When y=1, x=.851 --- When y=-1, x=1.111

|1-0.851| = 0.149
|1-1.111| = 0.111

Delta = 0.111 ?[/STRIKE]


2. I'm not really sure how to solve this...

 

[mighty2000]

Hi there!

For the limit problem, it looks like you have the right idea. However, there are a few things that could use some clarification. First, you correctly identified that ε = 1, but the value of δ that you are solving for is the distance between x and 1, not the value of x itself. So, when you have y = 1, the value of x should be 0.851, not the other way around. Similarly, when y = -1, the value of x should be 1.111. So, your calculations for the delta values are actually correct, but make sure to use the correct values of x.

For the second problem, it looks like you are being asked to find a value of δ that satisfies the condition that the difference between f(x) and 1 is less than 0.5. In other words, we want to find a value of δ such that |f(x) - 1| < 0.5. Looking at the graph of f(x) = 1/x, we can see that as x approaches 0, the value of f(x) approaches infinity. So, we need to choose a value of δ that is close enough to 0 such that the value of f(x) is less than 0.5 away from 1. Based on the graph, it looks like a good choice for δ would be 0.1. This means that for all values of x that are within 0.1 of 0 (i.e. between -0.1 and 0.1), the value of f(x) should be within 0.5 of 1. I hope this helps! Good luck on your test tomorrow!