   SEARCH HOME Math Central Quandaries & Queries  Question from Jackie: how to evaluate limit sinx/x as x tends to zero if x is in degrees Jackie, if x is in degrees, then let y = x(2π / 360). Then y is in radians.

So limit sin (x degrees) / x as x tends to zero = limit sin (y / (2π / 360) / [y / (2π / 360) ] as y tends to zero.

Hope this helps,
Stephen La Rocque.

Jackie wrote back

dear Stephen La Rocque.,
thanks very much for your kind hint to my problem but this is very part I got into difficulty and
I would very grateful if you can show me how to evaluate this expression

limit sin (x degrees) / x as x tends to zero = limit sin (y / (2π / 360) / [y / (2π / 360) ] as y tends to zero.

Thanks
Jackie

Hi Jackie,

There are two sine functions in this problem. You know them both because they are both on your calculator. One of them returns the sine of an angle if you input the measure of the angle in degrees and the other returns the sine of an angle if you input the measure of the angle in radians. We usually refer to both of these functions by sin(t) because in a specific problem we know whether we are working in degrees or radians. In this problem you are dealing with both functions so I want to distinguish between them.

• Let sin(t) be the function that returns the sine of an angle if t is the measure of the angle in degrees.

• Let SIN(t) be the function that returns the sine of an angle if t is the measure of the angle in radians.

What you know from your calculus class is that limit SIN(t)/t approaches 1 as t approaches zero.

As Stephen pointed out, if x is the measure of an angle in degrees then y is the measure of the angle in radians if y = x(2π / 360). In this situation sin(x) = SIN(y), it's the same angle you have just used different units to measure it.

Thus sin(x)/x = SIN (y)/[y / (2π / 360) ] = (SIN(y)/y) × π/180. Finally as x approaches zero so does y and hence sin(x)/x approaches 1 × π/180 = π/180.

I hope this helps,
Harley     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.