0 Comments
Answer:
To find the limit of the function as x approaches infinity, we can use L'Hôpital's rule. Let's apply the rule:
lim x→∞ x sin(3/x)
We can rewrite this expression as:
lim x→∞ (sin(3/x))/(1/x)
Now, we can differentiate the numerator and denominator separately. Applying L'Hôpital's rule:
lim x→∞ (cos(3/x) * (-3/x^2))/(-1/x^2)
Simplifying further:
lim x→∞ (3cos(3/x))/1
Now, as x approaches infinity, the term 3cos(3/x) approaches 3cos(0) = 3.
Therefore, the limit is:
lim x→∞ (3cos(3/x))/1 = 3
So, the limit of x sin(3/x) as x approaches infinity is 3.
To find the limit , we can use L'Hôpital's rule.
Applying L'Hôpital's rule, we differentiate the numerator and the denominator separately. Let's start by differentiating the numerator.
Differentiating with respect to gives .
Now, let's differentiate the denominator.
Differentiating with respect to requires the chain rule. The derivative of with respect to is . So, the derivative of with respect to is .
Taking the limit of as approaches infinity, we obtain .
Therefore, .
Note that in this case, we can also use an elementary method without L'Hôpital's rule. Since , we can substitute and rewrite the limit as . As and , the limit is also .
9.4m questions
12.2m answers