No.Michele Nunes said:Homework Statement
lim (x)(sin(1/x))
x->∞
Homework Equations
The Attempt at a Solution
The correct answer is 1, however I do not understand why. I thought that 1/∞ is essentially 0 and sin(0) = 0 so the whole limit would be 0.
Michele Nunes said:When I graphed it, the limit did seem to approach 1 though. I don't understand why it doesn't work out algebraically.
Ohh, I wasn't aware that 0 * ∞ is considered indeterminate form as well, but that's good to know now though, thank you!Mark44 said:No.
You have x becoming unbounded while sin(1/x) is approaching 0. This is the indeterminate form [0 * ∞], meaning that we can't say without taking the limits what will happen.
The limit as X approaches infinity of (X)(sin(1/X)) is equal to 1. This can be found by using L'Hospital's rule and taking the derivative of both the numerator and denominator, which gives us (sin(1/X))/(1/X^2). As X approaches infinity, the term 1/X^2 becomes negligible and we are left with the limit of sin(1/X) which is equal to 1.
The limit of (X)(sin(1/X)) approaches 1 as X approaches infinity because the term 1/X becomes increasingly smaller as X gets larger. This causes the term sin(1/X) to also become smaller and approach 0. As a result, we are left with the limit of 1 which is equal to 1.
The limit as X approaches infinity of (X)(sin(1/X)) is significant because it shows that even though the function (X)(sin(1/X)) may oscillate and have multiple values for different X values, as X approaches infinity, the function approaches a single, finite value of 1. This concept is important in calculus and helps us understand the behavior of functions as their inputs get larger and larger.
The limit as X approaches infinity of (X)(sin(1/X)) can be found by using L'Hospital's rule, which states that the limit of a quotient of two functions is equal to the limit of their derivatives. By taking the derivative of both the numerator and denominator, we can simplify the expression and find the limit to be 1.
No, the limit as X approaches infinity of (X)(sin(1/X)) is only equal to 1 for X values approaching infinity. For all other values of X, the function will have a different value. For example, when X=1, the function is equal to sin(1/1) which is approximately 0.84. However, as X gets larger and larger, the function will approach a limit of 1.