limit exponential function equation example with solution pleasee!
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limit exponential function equation example with solution pleasee!
limit exponential function equation example with solution pleasee!
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Answer:
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Step-by-step explanation:
A special type of exponential function appears frequently in real-world applications. To describe it, consider the following example of exponential growth, which arises from compounding interest in a savings account. Suppose a person invests P dollars in a savings account with an annual interest rate r , compounded annually. The amount of money after 1 year is
A(1)=P+rP=P(1+r) .
The amount of money after 2 years is
A(2)=A(1)+rA(1)=P(1+r)+rP(1+r)=P(1+r)2 .
More generally, the amount after t years is
A(t)=P(1+r)t .
If the money is compounded 2 times per year, the amount of money after half a year is
A(12)=P+(r2)P=P(1+(r2)) .
The amount of money after 1 year is
A(1)=A(12)+(r2)A(12)=P(1+r2)+r2(P(1+r2))=P(1+r2)2.
After t years, the amount of money in the account is
A(t)=P(1+r2)2t .
More generally, if the money is compounded n times per year, the amount of money in the account after t years is given by the function
A(t)=P(1+rn)nt.
What happens as n→∞? To answer this question, we let m=n/r and write
(1+rn)nt=(1+1m)mrt,
and examine the behavior of (1+1/m)m as m→∞ , using a table of values (Table).
Values of (1+1m)m as m→∞