To find the limit of the expression as h approaches 0, we can use the concept of trigonometric limits. Let's break down the expression step by step to understand how the limit evaluates to 0.



The given expression is hsin(1/h). To simplify it further, we can rewrite it as sin(1/h)/h. This step allows us to work with a more manageable form of the expression.



To evaluate the limit, we need to determine the behavior of sin(1/h)/h as h approaches 0. Trigonometric limits tell us that the limit of sin(x)/x as x approaches 0 is equal to 1.


Trigonometric Limit: lim(x→0) sin(x)/x = 1

In our expression, we have sin(1/h)/h. By substituting x = 1/h, we can rewrite the expression as sin(x)/x. Therefore, the limit of hsin(1/h) as h approaches 0 is equivalent to sin(x)/x as x approaches infinity.



Using the trigonometric limit mentioned above, we can conclude that sin(x)/x approaches 1 as x approaches 0. Since h approaches 0, we can substitute x = 1/h back into the expression.

Therefore, sin(x)/x = sin(1/h)/h approaches 1 as h approaches 0.


Trigonometric Limit: lim(x→0) sin(x)/x = 1



By applying the trigonometric limit, we have established that the limit of hsin(1/h) as h approaches 0 is equal to 1. Therefore, the given expression, hsin(1/h), evaluates to 0 as h tends to 0.


The limit of hsin(1/h) as h approaches 0 is 0.

Note: It is important to note that this limit does not imply that hsin(1/h) is equal to 0 for all values of h. The limit only gives us information about the behavior of the expression as h approaches 0.