How would you go about answering this?
If x and y are both positive evaluate
A.) lim= (ln(0.8x^p+2.0y^p))/p
p->0
B.) lim= (0.8x^p+2.0y^p)^(1/p)
p->0
For part A, we can use L'Hopital's rule:
lim [ln(0.8x^p + 2.0y^p)]/p
p->0
= lim [(0.8x^p + 2.0y^p)/(ln(0.8x^p + 2.0y^p))^-1]
p->0
Now, we can use L'Hopital's rule to evaluate the limit of the denominator:
= lim [(0.8x^p + 2.0y^p)/(-1)(0.8x^p + 2.0y^p)^(-2)(0.8ln(0.8x^p + 2.0y^p) + 2.0pyp^-1))]
p->0
= lim [-0.8ln(0.8x^p + 2.0y^p)/(0.8x^p + 2.0y^p)]
p->0
= -ln(0.8)/(x^0 + y^0)
(since lim (0.8x^p + 2.0y^p) = 0.8x^0 + 2.0y^0 = 2 > 0)
= -ln(0.8)/(x + y)
Therefore, the answer for part A is -ln(0.8)/(x + y).
For part B, we can rewrite the expression as:
lim [0.8x^p + 2.0y^p]^(1/p)
p->0
= lim e^(ln(0.8x^p + 2.0y^p)/p)
p->0
Now, we can use L'Hopital's rule again:
= lim e^[(ln(0.8x^p + 2.0y^p))^-1(0.8x^p ln(0.8x^p + 2.0y^p) + 2.0y^p ln(0.8x^p + 2.0y^p))]
p->0
= e^(0.8ln(0.8) + 2.0ln(2))
(since lim (0.8x^p + 2.0y^p) = 0.8x^0 + 2.0y^0 = 2 > 0)
= e^(ln(0.8^0.8 * 2^2))
= e^(ln(1.024))
= 1.024
Therefore, the answer for part B is 1.024.
I don't think the limit exists. Consider the case where y=0. Then
ln(0.8x^p)/p = (ln0.8 + p*lnx)/p = ln0.8/p + lnx
As p→0, lnx→-∞ and 0.8/p→∞
And if y is not zero, then you have yet another number divided by p, which will also be undefined.
You are correct. I apologize for my mistake in the previous answer. As you have pointed out, if y=0, then the limit does not exist. Additionally, if y is not zero, then we cannot apply L'Hopital's rule to evaluate the limit. Therefore, the limit is undefined in both parts A and B. Thank you for bringing this to my attention.
To evaluate the limits A) and B) as p approaches 0, we can use the technique of L'Hôpital's Rule, which states that if an indeterminate form (such as 0/0 or ∞/∞) is obtained when evaluating a limit, the ratio of the derivatives of the numerator and denominator can be taken instead.
Let's calculate each limit using L'Hôpital's Rule separately:
A) Evaluating lim as p approaches 0: (ln(0.8x^p+2.0y^p))/p
1. Take the derivative of the numerator: d/dp(ln(0.8x^p+2.0y^p)) = (0.8x^p*ln(0.8x^p+2.0y^p))/(0.8x^p+2.0y^p)
2. Take the derivative of the denominator: d/dp(p) = 1
3. The limit becomes lim as p approaches 0: ((0.8x^p*ln(0.8x^p+2.0y^p))/(0.8x^p+2.0y^p))/1
= (0.8x^p*ln(0.8x^p+2.0y^p))/(0.8x^p+2.0y^p)
B) Evaluating lim as p approaches 0: (0.8x^p+2.0y^p)^(1/p)
1. Take the derivative of the exponent: d/dp(1/p) = -1/p^2
2. Multiply the derivative by the original expression: (0.8x^p+2.0y^p)*(-1/p^2) = -(0.8x^p+2.0y^p)/(p^2)
3. Take the limit of the new expression: lim as p approaches 0: -(0.8x^p+2.0y^p)/(p^2)
After applying L'Hôpital's Rule, we have simplified both limits A) and B). Now, in order to evaluate the limits, we need the specific values of x and y.