Actuarial Outpost What is the limit?
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#1
05-25-2018, 01:07 PM
 ericp Member Join Date: Aug 2007 Posts: 283
What is the limit?

Anyone know how to evaluate this limit?

(10,000 / (x+10,000))^2
Y = divided by
(x goes to infinity) (20,000 / (20,000 +x^2)

The answer is 10,000^2 / 20,000. I don't get that answer. The solution did not show method.

Switching terms around I have
[10,000^2/20,000] * [(x+10,000)^2 /(20,000 + x^2)]. If the right side was 0/0 i would take the derivative of numerator and denominator and try again but I don't think that works for infinity divided by infinity.

Anyone out there know why the right side end up being one?
#2
05-25-2018, 04:16 PM
 SIGUS Member CAS SOA Join Date: Oct 2014 Posts: 50

0/0 and infinity/infinity are both indeterminate limit forms so the l'hopital's rule can be applied in both situations
#3
05-25-2018, 06:33 PM
 Abraham Weishaus Member SOA AAA Join Date: Oct 2001 Posts: 7,197

Quote:
 Originally Posted by ericp Anyone know how to evaluate this limit? (10,000 / (x+10,000))^2 Y = divided by (x goes to infinity) (20,000 / (20,000 +x^2) The answer is 10,000^2 / 20,000. I don't get that answer. The solution did not show method. Switching terms around I have [10,000^2/20,000] * [(x+10,000)^2 /(20,000 + x^2)]. If the right side was 0/0 i would take the derivative of numerator and denominator and try again but I don't think that works for infinity divided by infinity. Anyone out there know why the right side end up being one?
You can take derivatives twice and use L'Hospital. Probably simpler is noticing that the numerator and denominator are both quadratic polynomials, and the x^2 term in each dominates, so the limit is the coefficient of x^2 in numerator divided by coefficient of x^2 in denominator.
#4
05-30-2018, 01:07 PM
 ericp Member Join Date: Aug 2007 Posts: 283

Thanks for responses. I did not realize that both were indeterminate forms and forgot that the highest term in numerator and denominator dominate for limits.