There is no limit. As x approaches 0, sin 1/x oscillates between -1 and 1 infinitely many times. This is considered a form of divergence.
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You can use the L'hopital's rule to calculate the limit of e5x -1 divided by sin x as x approaches 0.
As X approaches infinity it approaches close as you like to 0. so, sin(-1/2)
The limit of cos2(x)/x as x approaches 0 does not exist. As x approaches 0 from the left, the limit is negative infinity. As x approaches 0 from the right, the limit is positive infinity. These two values would have to be equal for a limit to exist.
When the limit as the function approaches from the left, doesn't equal the limit as the function approaches from the right. For example, let's look at the function 1/x as x approaches 0. As it approaches 0 from the left, it travels towards negative infinity. As it approaches 0 from the right, it travels towards positive infinity. Therefore, the limit of the function as it approaches 0 does not exist.
The limit is 0.