- #1
lizzie
- 25
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find lim (x^2n - 1)/(x^2n + 1)
x->infinity
-> means tends to
x->infinity
-> means tends to
lizzie said:find lim (x^2n - 1)/(x^2n + 1)
x->infinity
-> means tends to
lizzie said:find lim (x^2n - 1)/(x^2n + 1)
x->infinity
-> means tends to
DeaconJohn said:Off the top of my head, my best guess is ...
For real numbers, the limit is equal to 1 if x > 1, -1/2 if x = 1, -1 if 0<=x<1, and is undefined if x<0.
For complex numbers with non-zero imaginary part, the limit is equal to -1 if |x| < 1 and is undefined if |x| >= 1.
HallsofIvy said:You asked that originally and you have already been given 4 answers.
Gib Z said:The original question asked for x--> infinity, not n.
As for the original question- Try adding and then subtracting 2 off the numerator.
For when n --> infinity, DJ had an attempt but needs some corrections: the limit is equal to one if |x| > 1, 0 if |x|=1, -1 if |x| < 1 and not undefined for any values. Note that we have an even function, so none of the "undefined if x<0" stuff.
The limit of the function (x^2n - 1)/(x^2n + 1) as x approaches infinity is 1. As x gets larger and larger, the terms with higher exponents (x^2n) become insignificant compared to 1. Therefore, we can simplify the function to 1/1, which equals 1.
We can prove this limit by using the definition of a limit. Let L be the limit of the function as x approaches infinity. Then, for any positive number e, there exists a corresponding positive number N such that for all x greater than N, the absolute value of (x^2n - 1)/(x^2n + 1) - L is less than e. By simplifying the function to 1/1 and setting N to a large enough value, we can show that this condition is met, and therefore, the limit is 1.
Yes, the limit of the function (x^2n - 1)/(x^2n + 1) as x approaches infinity is always 1. This is because, as mentioned in the first answer, the terms with higher exponents become insignificant compared to 1 as x gets larger and larger. Therefore, the limit is always 1.
No, the limit of the function (x^2n - 1)/(x^2n + 1) as x approaches infinity cannot be negative. Since the terms with higher exponents become insignificant compared to 1, the numerator will always be 1 or a positive number, while the denominator will always be a positive number. Therefore, the limit will always be a positive number, and cannot be negative.
The value of n does not affect the limit of the function (x^2n - 1)/(x^2n + 1) as x approaches infinity. As n increases, the terms with higher exponents become even more insignificant compared to 1, but the overall limit remains 1. This is because, no matter the value of n, the function can always be simplified to 1/1, which equals 1.