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Mathematics Dictionary |
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Subject : Application of prgram ABH on internat |
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Keyword : Limit of (1+x)^(1/x) = e and x = ? |
* Q2. How to use the program ABH to find answer from graphs ?
2. Click a subject in upper box 3. Click a program in lower box 4. Click Back command and repeat step 3 or steps 2 and 3 * Q3. How to run program 01 03 ?
2. Click Subject 01 in upper box 3. Click program 03 in lower box * Q4. How to start the program ?
2. Select run at current location (Donot download) 3. Select yes to run |
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Q1 Prove that Limit of (1+x)^(1/x) = e when x goes to 0 A1. Graphic answer :
* Click run at current location * Click yes then enter program ABG * Click start and subjects loaded into upper box * Click subject 08 and programs loaded into lower box * Click program 04 to see graphs * Click back command to see other programs A1. Binomial Theorem method :
* Picture Mathematics by Dr K. G. Shih * Page 167-168 * Binomial theorem : (1+a)^n=1+n*a + n*(n-1)*a^2/2! + .... * Let a=1/x and n=x * We have (1+1/x)^x = 1 + x*(1/x) + x*(x-1)*(1/x)^2/2! + .... * When x = infinite x*(x-1)/(x^2) = 1 * Hence Limit of (1+1/x)^x = 1 + 1 + 1/2! + 1/3! + 1/4! + ... * Since exp(x) = 1 + x + x^2/2! + x^3/3! + x^4/4! + ...... * And exp(1) = e = 1 + 1 + 1/2! + 1/3! + .... * We get Limit of (1+1/x)^x = e when x goes to infinite * or Limit (1+u)^(1/u) = e when u goes to 0 (i.e. u=1/x) |
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Q2. Limit of (1+a/(b*x))^(c*x) = ? when x = infinite A2. The solution is
* = Limit[(1+1/u)^u)]^(a*c/b) where u=b*x/a * = (a*c/b) |
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Q3 Find graphic solution of Limit[(1+1/(2*x))^x] as x=infinite A3. Software ABH program 08 07
* Click subject 08 Limit * Click program 07 Limit[(1+1/(a*x))^(b*x)] * Give input a = 2 and b=1 * The answer is e^(1/2) * Verify the following identities * Lim[(1+1/(2*x)^x] = exp(1/2) as x goes to infinite * Lim[(1+1/(3*x)^x] = exp(1/3) as x goes to infinite * Lim[(1+1/(4*x)^x] = exp(1/4) as x goes to infinite |
Q4 Verify following expression if x = infinite by ABH program 08 07
* Limit[(1+1/(x))^(2*x)] = e^2 * Limit[(1+1/(x))^(3*x)] = e^3 * Limit[(1+1/(x))^(4*x)] = e^4 |
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Reference |
| Show Room of MD2002 | Contact Dr. Shih | Math Examples Room |
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