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vktsn0303
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How can it be proved that as lim n tends to infinity, (n2-1)/(n2 + n + 1) tends to 1 ?
jbunniii said:Divide the numerator and denominator by ##n^2## to obtain
$$\lim_{n \to\infty} \frac{1 - 1/n^2}{1 + 1/n + 1/n^2}$$
The numerator and denominator both have limit ##1## as ##n \to\infty##, therefore so does the quotient.
vktsn0303 said:How can it be proved that as lim n tends to infinity, (n2-1)/(n2 + n + 1) tends to 1 ?
you are right.jbunniii said:Divide the numerator and denominator by ##n^2## to obtain
$$\lim_{n \to\infty} \frac{1 - 1/n^2}{1 + 1/n + 1/n^2}$$
The numerator and denominator both have limit ##1## as ##n \to\infty##, therefore so does the quotient.
Because that would be assuming that the limit of the quotient is the quotient of the limits. That principle is almost true. It holds when all three limits exist (and when the limit of the denominator is non-zero). But when the limit of the numerator and denominator both fail to exist, one is left with no prediction for the limit of the quotient. It is an "indeterminate form".vktsn0303 said:Well, I actually had a fundamental doubt (silly even) for which reason I had posted the question mainly. I am a newbie to the concept of limits. My doubt is as follows. Why should we look to modify the numerator or the denominator or both? Why not just consider that as n tends to infinity the division in question gives us (infinity)/(infinity) which is not defined.
You're not really modifying either the numerator or denominator. All that's happening is the factoring of both num. and denom.vktsn0303 said:Well, I actually had a fundamental doubt (silly even) for which reason I had posted the question mainly. I am a newbie to the concept of limits. My doubt is as follows. Why should we look to modify the numerator or the denominator or both?
vktsn0303 said:Why not just consider that as n tends to infinity the division in question gives us (infinity)/(infinity) which is not defined. This would lead us to the conclusion that the problem is therefore not solvable. Why can't this just be the case. I have actually observed many problems being solved by modifying the numerator and denominator. But it just seems so obvious that (infinity)/(infinity) is the actual result.
Consider a very simple example. Let us define a constant sequence: ##x_n = 1## for all positive integers ##n##. Clearly this sequence has limit ##1##. Now let's rewrite the same sequence another way. Most likely you agree that ##n/n = 1## for every ##n##, so I can write ##x_n = n/n##. I have not changed the value of the sequence for any ##n##, so its limit cannot change: it is still ##1##. It doesn't become ##\infty/\infty## just because the numerator and denominator both approach infinity.vktsn0303 said:Why should we look to modify the numerator or the denominator or both? Why not just consider that as n tends to infinity the division in question gives us (infinity)/(infinity) which is not defined. This would lead us to the conclusion that the problem is therefore not solvable. Why can't this just be the case. I have actually observed many problems being solved by modifying the numerator and denominator. But it just seems so obvious that (infinity)/(infinity) is the actual result.
vktsn0303 said:Well, I actually had a fundamental doubt (silly even) for which reason I had posted the question mainly. I am a newbie to the concept of limits. My doubt is as follows. Why should we look to modify the numerator or the denominator or both? Why not just consider that as n tends to infinity the division in question gives us (infinity)/(infinity) which is not defined. This would lead us to the conclusion that the problem is therefore not solvable. Why can't this just be the case. I have actually observed many problems being solved by modifying the numerator and denominator. But it just seems so obvious that (infinity)/(infinity) is the actual result.
The concept of limit in mathematics refers to the value that a function or sequence approaches as the input or variable approaches a certain value. It is used to describe the behavior of a function or sequence as the input or variable gets closer and closer to a specific value.
The limit of a function or sequence can be calculated using different techniques, such as algebraic manipulation, graphing, or using the limit laws. In the case of the limit n2-1/(n2 + n + 1)→1, the limit can be evaluated by simplifying the expression and plugging in the specific value that the variable is approaching.
The notation n2-1/(n2 + n + 1)→1 represents the limit of the function (n2-1)/(n2 + n + 1) as n approaches 1. This means that as the value of n gets closer and closer to 1, the value of the function also gets closer to the limit, which is 1.
Understanding the concept of limit is important in mathematics because it helps us to better understand the behavior of functions and sequences. It also allows us to solve more complex problems and make predictions about the behavior of a function or sequence as the input or variable approaches a specific value.
The concept of limit is used in various real-world applications, such as in physics, engineering, and economics. For example, it is used to calculate the speed of an object or the rate of change in a system. It is also used in finance to calculate the future value of an investment or to predict the growth of a population.