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March 27, 2022 at 10:49 am #9249
Making Use of an Online Limits Calculator to Show Six Standard Identities Currently Included in the Actuarial Life Contingencies Syllabus
Starting with the two equations:
1. e^(delta)=v and
2. (1d(m)/m)^m=v, show that lim m→infinity d(m)=delta.
Method
Rearranging equation 1. gives delta=ln(v) and rearranging equation 2. gives d(m)=m*(1v^(1/m)), meaning that lim m→infinity d(m)=lim m→infinity m*(1v^(1/m)), which, after using an Online Limits Calculator, is found to equal –ln(v) which equals delta.
Starting with the two equations:
1. e^(delta)=v and
2. (1+i(m)/m)^m=v^(1), show that lim m→infinity i(m)=delta.
Method
Rearranging equation 1. gives delta=ln(v) and rearranging equation 2. gives i(m)=m*(v^(1/m)1), meaning that lim m→infinity i(m)=lim m→infinity m*(v^(1/m)1), which, after using an Online Limits Calculator, is found to equal –ln(v) which equals delta.
Starting with the four equations:
1. 1+i=v^(1)
2. 1d=v
3. (1+i(m)/m)^m=v^(1) and
4. (1d(m)/m)^m=v, show that lim v→1 i*d/(i(m)*d(m))=1.
Method
Rearranging equation 1. gives i=v^(1)1, rearranging equation 2. gives d=1v, rearranging equation 3. gives i(m)=m*(v^(1/m)1) and rearranging equation 4. gives d(m)=m*(1v^(1/m)), meaning that lim v→1 i*d/(i(m)*d(m))=lim v→1 (v^(1)1)*(1v)/(m*(v^(1/m)1)*m*(1v^(1/m))), which, after using an Online Limits Calculator, is found to equal 1.
Starting with the three equations:
1. 1+i=v^(1)
2. (1+i(m)/m)^m=v^(1) and
3. (1d(m)/m)^m=v, show that lim v→1 (ii(m))/(i(m)*d(m))=(m1)/(2*m).
Method
Rearranging equation 1. gives i=v^(1)1, rearranging equation 2. gives i(m)=m*(v^(1/m)1) and rearranging equation 3. gives d(m)=m*(1v^(1/m)), meaning that lim v→1 (ii(m))/(i(m)*d(m))=lim v→1 (v^(1)1m*(v^(1/m)1))/(m*(v^(1/m)1)*m*(1v^(1/m))), which, after using an Online Limits Calculator, is found to equal (m1)/(2*m).
Starting with the three equations:
1. 1+i=v^(1)
2. 1d=v and
3. e^(delta)=v, show that lim v→1 i*d/delta^2=1.
Method
Rearranging equation 1. gives i=v^(1)1, rearranging equation 2. gives d=1v and rearranging equation 3. gives delta=ln(v), meaning that lim v→1 i*d/delta^2=lim v→1 (v^(1)1)*(1v)/(ln(v))^2, which, after using an Online Limits Calculator, is found to equal 1.
Starting with the two equations:
1. 1+i=v^(1) and
2. e^(delta)=v, show that lim v→1 (idelta)/delta^2=1/2.
Method
Rearranging equation 1. gives i=v^(1)1 and rearranging equation 2. gives delta=ln(v), meaning that lim v→1 (idelta)/delta^2=lim v→1 (v^(1)1+ln(v))/(ln(v))^2, which, after using an Online Limits Calculator, is found to equal 1/2.
I used the Online Limits Calculator at Limit of a Function Calculator – Online Limits Solver
Richard Purvey, March 2022.

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