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\[\mathop \limits_ \frac = \frac\mathop \limits_ \frac = \frac\left( 1 \right) = \frac\] Now, in this case we can’t factor the 6 out of the sine so we’re stuck with it there and we’ll need to figure out a way to deal with it.To do this problem we need to notice that in the fact the argument of the sine is the same as the denominator ( both \(\theta \)’s).
So, let \(\theta = x - 4\) and then notice that as \(x \to 4\) we have \(\theta \to 0\).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.They can’t always be done, but sometimes, such as this case, they can simplify the problem.The change of variables here is to let \(\theta = 6x\) and then notice that as \(x \to 0\) we also have \(\theta \to 6\left( 0 \right) = 0\).The formulas below would pick up an extra constant that would just get in the way of our work and so we use radians to avoid that.So, remember to always use radians in a Calculus class!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.If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.With this section we’re going to start looking at the derivatives of functions other than polynomials or roots of polynomials.