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The limit is 0.
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.
You can use the L'hopital's rule to calculate the limit of e5x -1 divided by sin x as x approaches 0.
limit x tends to infinitive ((e^x)-1)/(x)