Find the limit. Use l'Hospital's Rule if appropriate. If there is a more elementary method, consider using it.
lim x→∞ (x tan(3/x))
let u = 1/x
then you have
tan(3u)/u as u->0
we know that tan(u)/u -> 0, so
tan(3u)/u -> 3
To find the limit of the function as x approaches infinity, we can first simplify the expression. We have:
lim x→∞ (x tan(3/x))
Since tan(3/x) approaches 0 as x approaches infinity, we can rewrite the expression as:
lim x→∞ (x)(0)
Any number multiplied by 0 is equal to 0. Therefore, the limit of the function as x approaches infinity is 0.
Now let's explain how we arrived at this conclusion:
1. Rewrite the expression: First, we rewrite the given expression as x times the tangent of 3/x. This is the same as x * tan(3/x).
2. Simplify the expression: As x approaches infinity, 3/x approaches 0. We can use this fact to simplify the expression.
3. Apply the limit rule: Since tan(0) is equal to 0, and x approaches infinity, we can rewrite the expression as (x)(0), which is 0.
4. Evaluate the limit: Any number multiplied by 0 is equal to 0. Therefore, the limit of the function as x approaches infinity is 0.
In this case, we didn't need to use l'Hospital's Rule or any other advanced method, as the limit was easily derived through simplification and basic properties of limits.