# Limit as X approaches ∞ of (X)(sin(1/X))

• Michele Nunes
In summary, the limit of (x)(sin(1/x)) as x approaches infinity is an indeterminate form of [0 * ∞], meaning that we cannot determine the result without taking the limits. While graphing the function may suggest the limit approaches 1, algebraically it is not possible to determine the exact value.
Michele Nunes

## Homework Statement

lim (x)(sin(1/x))
x->∞

## 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. When I graphed it, the limit did seem to approach 1 though. I don't understand why it doesn't work out algebraically.

UPDATE: wait never mind, i just realized how you can do it algebraically

Michele Nunes said:

## Homework Statement

lim (x)(sin(1/x))
x->∞

## 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.
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.
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.

Michele Nunes
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.
Ohh, I wasn't aware that 0 * ∞ is considered indeterminate form as well, but that's good to know now though, thank you!

## 1. What is the limit as X approaches infinity of (X)(sin(1/X))?

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.

## 2. Why does the limit of (X)(sin(1/X)) approach 1 as X approaches infinity?

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.

## 3. Can you explain the significance of the limit as X approaches infinity of (X)(sin(1/X))?

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.

## 4. How do you find the limit as X approaches infinity of (X)(sin(1/X))?

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.

## 5. Is the limit as X approaches infinity of (X)(sin(1/X)) equal to 1 for all values of X?

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.

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