What is the limit of (1+1/x)^(x+1/x) as x -> infinity ?

This is an interesting limit. Lets define a variable y as:
y=lim_(x->oo) (1+1/x)^(x+1/x)
Lets take the logarithm of both sides.
ln(y)=lim_(x->oo) ln[(1+1/x)^(x+1/x)]
ln(y)=lim_(x->oo) (x+1/x) ln(1+1/x)
Now I am going to make a quick substitution in the logarithm on the right hand side.
u=1/x
ln(y)=lim_(x->oo) (x+1/x) ln(1+u)
Now I will use a common taylor expansion, ln(1+u)=sum_(n=0)^oo (-1)^(n-1)*(u^n)/n=u-1/2u^2+1/3u^3-...
ln(y)=lim_(x->oo) (x+1/x) ln(1+u)
ln(y)=lim_(x->oo) (x+1/x) (u-1/2u^2+1/3u^3-...)
Back substitute in to get in terms of x.
ln(y)=lim_(x->oo) (x+1/x) (1/x-1/(2x^2)+1/(3x^3)-...)
ln(y)=lim_(x->oo) [x (1/x-1/(2x^2)+1/(3x^3)-...)+1/x(1/x-1/(2x^2)+1/(3x^3)-...)]
ln(y)=lim_(x->oo) [(1-1/(2x)+1/(3x^2)-...)+(1/x^2-1/(2x^3)+1/(3x^4)-...)]
Now take the limit, every term that has an x in the denominator will go to zero leaving just one term.
ln(y)=1
Give both sides a new base e
e^ln(y)=e^1
y=e
Remember our original definition of y.
y=lim_(x->oo) (1+1/x)^(x+1/x)=e
This limit is equal to e.

Comments

Post a Comment

Popular posts from this blog

Single Variable Calculus, Chapter 3, 3.6, Section 3.6, Problem 34

Determine $y''$ of $\sqrt{x} + \sqrt{y} = 1$ by using implicit differentiation. Solving for 1st Derivative $ \begin{equation} \begin{aligned} \frac{d}{dx} (\sqrt{x}) + \frac{d}{dx} (\sqrt{y}) =& \frac{d}{dx} (1) \\ \\ \frac{d}{dx} (x)^{\frac{1}{2}} + \frac{d}{dx} (y)^{\frac{1}{2}} =& \frac{d}{dx} (1) \\ \\ \frac{1}{2} (x)^{\frac{-1}{2}} + \frac{1}{2} (y)^{\frac{-1}{2}} \frac{d}{dx} =& 0 \\ \\ \frac{1}{2 (y)^{\frac{1}{2}}} \frac{dy}{dx} =& \frac{-1}{2(x)^{\frac{1}{2}}} \\ \\ \frac{dy}{dx} =& - \frac{\cancel{2} (y)^{\frac{1}{2}}}{\cancel{2}(x)^{\frac{1}{2}}} \\ \\ \frac{dy}{dx} =& - \frac{(y)^{\frac{1}{2}}}{(x)^{\frac{1}{2}}} \end{aligned} \end{equation} $ Solving for 2nd Derivative $ \begin{equation} \begin{aligned} \frac{d^2 y}{dx^2} =& - \frac{\displaystyle (x)^{\frac{1}{2}} \frac{d}{dx}(y)^{\frac{1}{2}} - (y)^{\frac{1}{2}} \frac{d}{dx} (x)^{\frac{1}{2}} }{[(x)^{\frac{1}{2}}]^2} \\ \\ \\ \\ \frac{d^2 y}{dx^2} =& - \frac{\displaystyle (x)^{\frac{...
Read more

In “Fahrenheit 451,” what does Faber mean by “Those who don’t build must burn. It’s as old as history and juvenile delinquents”?

During a conversation between Montag and Faber, Faber says, "Those who don't build must burn. It's as old as history and juvenile delinquents" (Bradbury, 42). Faber is saying that those individuals who do not positively contribute to society are helping to destroy it. Faber's analogy corresponds to the function of the firefighters, who destroy books and prevent individuals from attaining knowledge. Montag, who is a jaded firefighter, feels extremely guilty about participating in the destruction of society. He visits Faber in hopes of altering the trajectory of his life and wishes to contribute to society instead of being a destructive force. Faber believes that authors, artists, and intellectuals "build" institutions and have an overall positive impact on society. According to Faber's views, individuals who do not give back to society either directly or indirectly contribute to its destruction, which explains his own feelings of guilt. While Faber do...
Read more

In what ways might RFID technology be used to serve customers better? What problems might arise? Do you think that the technology might be valuable when implanted in animals or people?

RFID stands for Radio Frequency Identification. It is a type of wireless technology and basically has three components to it: a tag, a reader, and a computer system. The tag consists of a microchip and radio antenna. The microchip in the tag contains essential information about a product or item. To transmit this information to a reader, the tag uses radio signals. After picking up the signals, the reader delivers the information to a computer system. From the computer system, companies can easily track the kinds of products consumers like to buy. This allows companies to position advertisements, sales, and product placements in stores according to their customer's preferences. Also, using RFID technology increases productivity: it is much easier to use, is more accurate, and less error-prone than traditional bar-coding. Thus, increased productivity leads to greater profits for companies. As for consumers, the benefits of RFID technology are numerous. For example, it minimizes wait...
Read more