lim x->2 (((1/(sqrt(2+s))-1)/(s+1))
mismatched parens. If that's
(1/√(2+s)-1) / (s+1)
then we have (1/√4 - 1)/3 = -1/6
if I got it wrong, fix it, but there's still no problem, as no denominator is zero.
The answer to the equation is -1/2. The parens are correct. But I was wrong on the limit. It is lim s->-1
bzzt. There are six ( and only 5 )
at any rate, l'Hospital's Rule will help, since the derivative of (s+1) is 1, and the zero denominator goes away.
Im confused on the "There are six (and only 5)" part. and How would you solve it without the I'Hospital's Rule? We haven't learned that yet.
six left parentheses, and only 5 right parentheses.
where you put that other paren will make a difference.
To find the limit as x approaches 2 of the expression (((1/(sqrt(2+s))-1)/(s+1))), we can use algebraic manipulation and the limit properties. Here's the step-by-step process:
Step 1: Simplify the expression
(((1/(sqrt(2+s))-1)/(s+1))) can be simplified as follows:
= ((1/(sqrt(2+s))-1)/(s+1))
= (((1 - sqrt(2+s))/(sqrt(2+s)))/(s+1))
= ((1 - sqrt(2+s)) / ((s+1) * (sqrt(2+s))))
Step 2: Factor out (s - 2) from the denominator
Since we are interested in the limit as x approaches 2, we can factor out (s - 2) from the denominator to simplify further. This will allow us to cancel out the factor of (s - 2) and evaluate the limit:
= (1 - sqrt(2+s)) / ((s+1) * (sqrt(2+s)))
= (1 - sqrt(2 + 2 + (s-2))) / ((s+1) * (sqrt(2+s)))
= (1 - sqrt(4 + (s-2))) / ((s+1) * (sqrt(2+s)))
= (1 - sqrt((s-2)^2 + 4)) / ((s+1) * (sqrt(2+s)))
Step 3: Evaluate the limit
Now that the expression is simplified, we can take the limit as s approaches 2:
lim (s->2) ((1 - sqrt((s-2)^2 + 4)) / ((s+1) * (sqrt(2+s))))
To find this limit, we can evaluate the expressions in the numerator and the denominator separately:
Numerator:
lim (s->2) (1 - sqrt((s-2)^2 + 4))
= 1 - sqrt((2-2)^2 + 4)
= 1 - sqrt(4)
= 1 - 2
= -1
Denominator:
lim (s->2) ((s+1) * (sqrt(2+s)))
= (2+1) * (sqrt(2+2))
= 3 * (sqrt(4))
= 3 * 2
= 6
Therefore, the final answer is:
lim (s->2) (((1/(sqrt(2+s))-1)/(s+1))) = -1/6.