Hey, we covered limits at the beginning of the year, but I cannot remember how to do them at all.
>>>what is the limit, as x approaches 2, of ((1/x )-(1/2))/(x-2)?
Please go step by step!
f= (1/x-.5)/(x-2)
as x>>2, (1/2=1/2)/(2-2) so it is of the form 0/0, so use LHopitals rule
as x>>2= (-1/x^2)/(1)= -1/4 / 1= -1/4
Sure! I'll walk you through the process step by step.
To find the limit as x approaches 2 of the expression ((1/x)-(1/2))/(x-2), we can simplify the expression by combining the fractions.
Step 1: Combine the fractions
The least common denominator (LCD) of the fractions 1/x and 1/2 is 2x. So we need to convert both fractions to have the same denominator.
(1/x) - (1/2) can be rewritten as ((2 - x)/(2x)).
Step 2: Simplify the expression
Now that the expression has a common denominator, we can simplify it further.
((2 - x)/(2x))/(x - 2) can be rewritten as ((2 - x)/(2x))/((x - 2)/1).
Step 3: Multiply by the reciprocal
To divide by a fraction, we can multiply by its reciprocal. So we'll multiply the whole expression by the reciprocal of the denominator, which is 1/(x - 2).
((2 - x)/(2x))/((x - 2)/1) * (1/(x - 2)) can be rewritten as ((2 - x)/(2x)) * (1/(x - 2)).
Step 4: Cancel out common factors
In the expression ((2 - x)/(2x)) * (1/(x - 2)), we can cancel out the common factor of (x - 2) in the numerator and denominator.
After canceling the common factor, we're left with ((2 - x)/(2x)) * (1/1), which simplifies to (2 - x)/(2x).
Step 5: Substitute the limit value
Now that we have simplified the expression, we can substitute the value x = 2 into the expression.
Substituting x = 2 into (2 - x)/(2x), we get (2 - 2)/(2 * 2) = 0/4 = 0.
Therefore, the limit as x approaches 2 of ((1/x)-(1/2))/(x-2) is 0.