Thread: Limits from first principles, L'Hopital's Rule - urgent help

Hi I would like to know if anyone can help me or guide me through the steps to the answers to these questions? I am fairy new to advanced mathematics and I am finding it a bit difficult to understand some topics...sigh... ur help would be greatly appreciated

1. Determine the limit from first prinicples by evaluating the function at values about the indicated limiting value.

lim [(x^3) - x]/(x-1)
x-->1


2. Evaluate;

lim [2- √(x+2)] / (x-2)
x-->2


3. A 5-Ω resistor and a variable resistor of resistance R are placed in parallel. The expression for the resulting resistance, Rτ, is given by
Rτ= 5R/ (5 +R). Determine the limiting value of Rτ as R-->.(Use L'Hopital's Rule)

1) Factor the numerator.

2) Rationalize the numerator

3) Divide numerator and denominator by the highest power variable (in this case x)

Originally Posted by Mythichaos

Hi I would like to know if anyone can help me or guide me through the steps to the answers to these questions? I am fairy new to advanced mathematics and I am finding it a bit difficult to understand some topics...sigh... ur help would be greatly appreciated

1. Determine the limit from first prinicples by evaluating the function at values about the indicated limiting value.

lim [(x^3) - x]/(x-1)
x-->1

The phrasing "by evaluating the function at values about the indicated limiting value" implies that you are expected to calculate for things like x= 0.9, 0.99, 0.999, 1.001, 1.01, 1.1 and then "guess" the limit. Of course, calculating any finite number of points won't actually tell you the limit for certain.

2. Evaluate;

lim [2- √(x+2)] / (x-2)
x-->2

Multiply both numerator and denominator by (rationalize the numerator).

3. A 5-Ω resistor and a variable resistor of resistance R are placed in parallel. The expression for the resulting resistance, Rτ, is given by
Rτ= 5R/ (5 +R). Determine the limiting value of Rτ as R-->.(Use L'Hopital's Rule)

What is the derivative of 5R with respect to R? What is the derivative of 5+ R with respect to R? Do you know what "L'Hopital's Rule" is?
Here, I agree with Chengben that L'Hopital's rule is not the simplest way to do this. (But the variable is "R", not "x"!)

ok.... for the second one.....
the answer :

lim [2- √(x+2)] / (x-2) = lim [2- √(x+2)] [2+ √(x+2)]/ (x-2) [2+ √(x+2)]
x-->2 x-->2

= lim [4 + 2√(x+2) - 2√(x+2) - (x+2)] / (x-2) [2+ √(x+2)]
x-->2

= lim (4 - x + 2) / (x-2) [2+ √(x+2)]
x-->2

.......then.... ?