See the Proof of Trig Limits section of the Extras chapter to see the proof of these two limits. Students often ask why we always use radians in a Calculus class. The proof of the formula involving sine above requires the angles to be in radians.
If the angles are in degrees the limit involving sine is not 1 and so the formulas we will derive below would also change.
When doing a change of variables in a limit we need to change all the \(x\)’s into \(\theta \)’s and that includes the one in the limit.
Doing the change of variables on this limit gives, \[\begin\mathop \limits_ \frac & = 6\mathop \limits_ \frac\hspace\theta = 6x\\ & = 6\mathop \limits_ \frac\\ & = 6\left( 1 \right)\\ & = 6\hspace\end\] And there we are.
Therefore, after doing the change of variable the limit becomes, \[\mathop \limits_ \frac = \mathop \limits_ \frac = 1\] The previous parts of this example all used the sine portion of the fact.
However, we could just have easily used the cosine portion so here is a quick example using the cosine portion to illustrate this.
A change of variables, in this case, is really only needed to make it clear that the fact does work.
In this case we appear to have a small problem in that the function we’re taking the limit of here is upside down compared to that in the fact.
\[\mathop \limits_ \frac = \mathop \limits_ \frac = 6\mathop \limits_ \frac\] Note that we factored the 6 in the numerator out of the limit.
At this point, while it may not look like it, we can use the fact above to finish the limit.