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# Calculus/Limits equal to e.

Question
Hi Randy,

I find using L'hospital rule and raising the powers to e^x then doing algebraic manipulation is too time consuming for solving limits like lim x---∞ (1+(3/x))^(x). Therefore, I have read about improper integrals as a possibility. I wonder can I use this method to more simply solve the above limit and can you briefly explain it or is there an even more simpler method.

Many Many Thanks,
Sam

Questioner:Sam
Category:Calculus
Private:No
Subject:Limits
Question:  Hi Randy,   <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<  ?????????????????

I find using L'hospital rule and raising the powers to e^x then doing algebraic manipulation is too time consuming for solving limits like lim x---∞ (1+(3/x))^(x). Therefore, I have read about improper integrals as a possibility. I wonder can I use this method to more simply solve the above limit and can you briefly explain it or is there an even more simpler method................

There is.  Write:  x = 3y

lim x---∞ (1+(3/x))^(x) =

lim y->∞ (1+(3/3y))^(3y)

lim y->∞ (1+1/y))^(3y)

lim y->∞ [(1+1/y))^y]^3

[lim y->∞ (1+1/y))^y] ^3

= e^3

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Calculus

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#### Paul Klarreich

##### Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

##### Experience

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

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