For the limit below, find values of δ that correspond to the ε values.
ε = .01 what is the δ rounded to five decimal places. Can anyone please explain to me how to do this problem? I know that the easiest way is to plug it into your graphing calculator and trace the line, but my TI-83 Plus is not accurate enough to give me any answer other than 1. Thank you!
What I would do is this: First determine that your function will be decreasing on the interval (3.99, 4.01) by taking the derivative and noting that it is negative. Then solve the equations and . The first will give you a value of x less than 1 and the second will give you a value of x greater than one. Since the function is decreasing on the interval (3.99, 4.01), the Extreme Value Theorem applies to say that these values of x yield the minimum and maximum values of f(x) on the interval. Now, you choose which of those values of x results in the smaller delta from 1.
Jhevon's method is probably easier than mine, since solving cubic equations is difficult. And yes, you would use Newton's method to approximate the answer to five decimal places of accuracy. Either that or plug it into your calculator's equation solver.
Here, I simply changed all the signs. (|a| = |-a|, it makes no difference). this was more for aesthetic reasons and ease of factoring, which I did in the next step
here is where i factor the left hand side of the previous step. i divided by and i got , and so i could factor it as you see.
how did i know to divide by x - 1? it is what we needed to factor out. it is our term. here,
...in retrospect, i would have actually done this a bit differently..., first letting |x - 1| < 1 or something as an initial delta bound.