Question

A) Use linear localization to makes plausible the fact that lim

x→0
​xsin(x)=1
(b) Find another way to confirm the result you found in part (a).

Answer 1

The question concerns calculating the limit of x multiplied by sine of x as x approaches 0. The correct answer is that the limit is 0, using linear localization and confirmation through another method such as L'Hôpital's rule or the bounded nature of the sine function.

Step-by-step explanation:

The student has asked a question about calculus, specifically regarding the limit of a function as the variable approaches a certain value. The question involves confirming that the limit of xsin(x) as x approaches 0 is indeed 1 using linear localization and then verifying the result by another method. The limit of xsin(x) as x tends to 0 is actually 0, not 1. This mistake could be a typo or misunderstanding on the student's part.

For part (a), we can use a linear approximation around 0 for the sine function, which gives us sin(x) ≈ x for small values of x. Therefore, the limit is:

lim x→0 x*sin(x) ≈ lim x→0 x*x = lim x→0 x^2 = 0.

For part (b), to confirm this result, we can employ L'Hôpital's rule or use the fact that sine of x is bounded between -1 and 1, which means xsin(x) will be limited by the magnitude of x, approaching 0 as x goes to 0.