How to Obtain limₓ͢ ₀ sin(x)/x by Algebraic Means

The demonstration of the fundamental trigonometric limit is generally found in Calculus I books, presented with geometric arguments.

We will show here an algebraic way of finding the same result, which in principle is already well known, that is,

This expression is not an identity, since it is only valid when the value of the variable tends to zero. However, if we subject it to a mathematical transformation, for example, if we set x=u+5, the expression remains valid, as long as this has repercussions on the form, see:

We will not demonstrate equality, we will therefore calculate the limit.

To do this, we will use the following transformation

And, in particular, we will choose the transformation

Which, consequently, leads us to write

See that first:

• x > u, in the range 0 to π/2;
• x = u, for x = 0.

So it can be deduced that:

dividing both sides by x, we have

and let’s rewrite this as

Such an expression will remain valid even if we delete the second term on the right-hand side, and in doing so we have:

Taking this expression as a basis, it will be valid to say that:

This expression will not be completely valid if we apply the transformation x=u, and once this is done, it must be rewritten just for this transformation, that is,

On this equality we will perform some algebraic operations according to the following steps.

This last expression will continue to be valid if we replace the index variable in the term of the second limit, from u to x tending to zero. And by doing this we have:

And finally, as the relationship sin(x)/x is an even function, we must obtain the same result when x belongs to the range - π/2 to 0. Therefore, it can be concluded that: