MHB Limits of m/n as x Approaches 1

  • Thread starter Thread starter spartas
  • Start date Start date
  • Tags Tags
    Limits
AI Thread Summary
The limit of the expression as x approaches 1 is derived using the formula for the sum of a geometric series. The limit can be rewritten to eliminate indeterminate forms by factoring out (x-1) from both the numerator and denominator. This leads to the simplified limit expression, which evaluates to the ratio of the sums of constants, resulting in m/n. The calculations confirm that as x approaches 1, the limit converges to m/n, where m and n are natural numbers. This demonstrates the relationship between the powers of x in the limit.
spartas
Messages
7
Reaction score
0
lim xm-1/xn-1 m,n elements of N
x→1
the answer is m/n but i have no idea how to start or solve this!
 
Mathematics news on Phys.org
Consider the following:

$$\sum_{k=0}^n\left(x^k\right)=\frac{x^{n+1}-1}{x-1}$$

Can you now rewrite the limit to get a determinate form?
 
We are given to find:

$$L=\lim_{x\to1}\frac{x^m-1}{x^n-1}$$ where $$m,n\in\mathbb{N}$$

Using the hint I suggested, we may write:

$$x^m-1=(x-1)\sum_{k=0}^{m-1}\left(x^k\right)$$

$$x^n-1=(x-1)\sum_{k=0}^{n-1}\left(x^k\right)$$

And so our limit becomes:

$$L=\lim_{x\to1}\frac{(x-1)\sum\limits_{k=0}^{m-1}\left(x^k\right)}{(x-1)\sum\limits_{k=0}^{n-1}\left(x^k\right)}=\lim_{x\to1}\frac{\sum\limits_{k=0}^{m-1}\left(x^k\right)}{\sum\limits_{k=0}^{n-1}\left(x^k\right)}=\frac{\sum\limits_{k=0}^{m-1}\left(1\right)}{\sum\limits_{k=0}^{n-1}\left(1\right)}=\frac{\sum\limits_{k=1}^{m}\left(1\right)}{\sum\limits_{k=1}^{n}\left(1\right)}=\frac{m}{n}$$
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
Back
Top