What is the limit as x approaches zero of sin(2x)/x

Answer 1

Beginning with the fact that the limit as x approaches zero of sin(x)/x is 1, we might infer,, when x goes to zero that 2x also goes to zero that the answer to the problem is 1. This is incorrect.

A good way to develop the correct answer is to replace sin(2x) with its identity 2sin(x)cos(x)

The problem now becomes the limit as x approaches zero of (2sin(x)cos(x))/x .

Since the limit of a product is the product of the limits this limit can be re written as the limit of

2cos(x) * sin(x)/x . The limit of this product would be the limit of 2cos(x) which is 2 times the limit of sin(x)/x which is 1.

The limit is then 2*1 = 2

While this is a good demonstration of the reasoning, limit problems similar to this one can be done more quickly by making the generalization that

the limit as x approaches 0 of sin(ax)/(ax) is 1 (True fact) #

Applying this fact to an example - Find the limit as x approaches 0 of sin(5x)/x . Unfortunately the problem as presented does not fit the form describe above in #.

We can make it fit the problem format by giving the denominator a coefficient of 5. This would change the problem, unless we also add a multiplier of 5 to the numerator. As a result of these moves, we will change the way the problem looks but not the problem itself.

This now gives us the original problem in this form - limit as x approaches zero of (5*sin(5x)/(5x)). The limit can then be identified as the constant 5 times the limit of sin(5x)/(5x) (equal to 1) thus

the answer is 5*1 = 5

end.