Calculus Tutorial 2.2.017

Problem Statement

eval rand(2,8)

eval pow(7,2)

eval 49+0.002

eval 49+0.001

eval 49+0.0005

eval 49+0.0001

eval round(((sqrt(49.002)-7)/(49.002-49)),7)

eval round(((sqrt(49.001)-7)/(49.001-49)),7)

eval round(((sqrt(49.0005)-7)/(49.0005-49)),7)

eval round(((sqrt(49.0001)-7)/(49.0001-49)),7)

eval round(0.0714285,4)

eval 49-0.002

eval 49-0.001

eval 49-0.0005

eval 49-0.0001

eval round(((sqrt(48.998)-7)/(48.998-49)),7)

eval round(((sqrt(48.999)-7)/(48.999-49)),7)

eval round(((sqrt(48.9995)-7)/(48.9995-49)),7)

eval round(((sqrt(48.9999)-7)/(48.9999-49)),7)

eval round(0.0714286,4)

eval 0.0714

Estimate the limit numerically or state that the limit doesn't exist.

Step 1

Question 1

To determine if a limit exists numerically for , construct a table of values of f(x) for x near c but greater than c (that is, ) and a second table of values of f(x) for x near c but less than c (that is, ). If both tables indicate convergence to the same number L, we take L to be an estimate for the limit.

In the given problem, c = 49 and

Complete the table of values of f(x) as . (Round your answers to seven decimal places.)

x	f(x)
49.002	0.0714278
49.001
49.0005
49.0001

As , f(x) approaches (rounded to four decimal places).

Great! Notice as the values of x get closer to 49, the results approach 0.0714.

You are evaluating the function for values of x that get closer to x = 49. Ask yourself if you see the results approach a number. Check your wrok carefully.

Incorrect.

Step 2

Question 2

Complete the table of values of f(x) as . (Round your answers to seven decimal places.)

x	f(x)
48.998	0.0714293
48.999
48.9995
48.9999

As , f(x) approaches (rounded to four decimal places).

Great! Notice as the values of x get closer to 49, the results approach 0.0714.

You are evaluating the function for values of x that get closer to x = 49. Ask yourself if you see the results approach a number. Check your wrok carefully.

Incorrect.

Step 3

Question 3

Since both tables indicate convergence to the same number 0.0714 as x approaches 49 from the left and from the right, then 0.0714 is an estimate for the limit and we write

Correct.

In both tables, ask yourself if as the value of x gets closer to c, the results approach a number. Is this the same number in both tables? If so, this number is the limit.

Incorrect.