Chapter 1. Calculus Tutorial 2.2.017

1.1 Problem Statement

eval rand(2,8)

eval pow($a,2)

eval $a2+0.002

eval $a2+0.001

eval $a2+0.0005

eval $a2+0.0001

eval round(((sqrt($a2plus002)-$a)/($a2plus002-$a2)),7)

eval round(((sqrt($a2plus001)-$a)/($a2plus001-$a2)),7)

eval round(((sqrt($a2plus0005)-$a)/($a2plus0005-$a2)),7)

eval round(((sqrt($a2plus0001)-$a)/($a2plus0001-$a2)),7)

eval round($fa2plus0001,4)

eval $a2-0.002

eval $a2-0.001

eval $a2-0.0005

eval $a2-0.0001

eval round(((sqrt($a2minus002)-$a)/($a2minus002-$a2)),7)

eval round(((sqrt($a2minus001)-$a)/($a2minus001-$a2)),7)

eval round(((sqrt($a2minus0005)-$a)/($a2minus0005-$a2)),7)

eval round(((sqrt($a2minus0001)-$a)/($a2minus0001-$a2)),7)

eval round($fa2minus0001,4)

eval $fa2minuslimit

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

1.2 Step 1

Question 1.1

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 = $a2 and

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

x	f(x)
$a2plus002	$fa2plus002
$a2plus001	lMbVGWePD7/xHvMovYTApQ==
$a2plus0005	+BXidD8JNd6iANd4/WFeNQ==
$a2plus0001	HdvB1pJvc99fzOFmntnzLA==

As , f(x) approaches qoxy4OxsqdTIsmPrKVDkRi1T0+E= (rounded to four decimal places).

Great! Notice as the values of x get closer to $a2, the results approach $fa2pluslimit.

You are evaluating the function for values of x that get closer to x = $a2. Ask yourself if you see the results approach a number. Check your wrok carefully.

Incorrect.

1.3 Step 2

Question 1.2

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

x	f(x)
$a2minus002	$fa2minus002
$a2minus001	JfC/0Fk+hCYzCXmaqF7FiA==
$a2minus0005	nn4zGX8rQf9MChuK6tZx8x91YUs=
$a2minus0001	oLtGUtLxc6KIb52j5dy12LrMozg=

As , f(x) approaches v8469tPPQiwl+nfqWJH7Au6Zii0= (rounded to four decimal places).

Great! Notice as the values of x get closer to $a2, the results approach $fa2minuslimit.

You are evaluating the function for values of x that get closer to x = $a2. Ask yourself if you see the results approach a number. Check your wrok carefully.

Incorrect.

1.4 Step 3

Question 1.3

Since both tables indicate convergence to the same number $fa2limit as x approaches $a2 from the left and from the right, then $fa2limit is an estimate for the limit and we write

= HJse94ng5dHcwsCi6O0ixw==

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.