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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. (1-d(m)/m)^m=v, show that lim m→infinity d(m)=delta.
Method
Re-arranging equation 1. gives delta=-ln(v) and re-arranging equation 2. gives d(m)=m*(1-v^(1/m)), meaning that lim m→infinity d(m)=lim m→infinity m*(1-v^(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
Re-arranging equation 1. gives delta=-ln(v) and re-arranging 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. 1-d=v
3. (1+i(m)/m)^m=v^(-1) and
4. (1-d(m)/m)^m=v, show that lim v→1 i*d/(i(m)*d(m))=1.
Method
Re-arranging equation 1. gives i=v^(-1)-1, re-arranging equation 2. gives d=1-v, re-arranging equation 3. gives i(m)=m*(v^(-1/m)-1) and re-arranging equation 4. gives d(m)=m*(1-v^(1/m)), meaning that lim v→1 i*d/(i(m)*d(m))=lim v→1 (v^(-1)-1)*(1-v)/(m*(v^(-1/m)-1)*m*(1-v^(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. (1-d(m)/m)^m=v, show that lim v→1 (i-i(m))/(i(m)*d(m))=(m-1)/(2*m).
Method
Re-arranging equation 1. gives i=v^(-1)-1, re-arranging equation 2. gives i(m)=m*(v^(-1/m)-1) and re-arranging equation 3. gives d(m)=m*(1-v^(1/m)), meaning that lim v→1 (i-i(m))/(i(m)*d(m))=lim v→1 (v^(-1)-1-m*(v^(-1/m)-1))/(m*(v^(-1/m)-1)*m*(1-v^(1/m))), which, after using an Online Limits Calculator, is found to equal (m-1)/(2*m).
Starting with the three equations:
1. 1+i=v^(-1)
2. 1-d=v and
3. e^(-delta)=v, show that lim v→1 i*d/delta^2=1.
Method
Re-arranging equation 1. gives i=v^(-1)-1, re-arranging equation 2. gives d=1-v and re-arranging equation 3. gives delta=-ln(v), meaning that lim v→1 i*d/delta^2=lim v→1 (v^(-1)-1)*(1-v)/(-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 (i-delta)/delta^2=1/2.
Method
Re-arranging equation 1. gives i=v^(-1)-1 and re-arranging equation 2. gives delta=-ln(v), meaning that lim v→1 (i-delta)/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
<p style=”text-align: left;”>Richard Purvey, March 2022.</p>If you take if far enough you might end up with a tool like this, which can write LTAM type questions on demand
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