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!This time we’re going to notice that it doesn’t really matter whether the sine is in the numerator or the denominator as long as the argument of the sine is the same as what’s in the numerator the limit is still one. \[\begin\mathop \limits_ \frac & = \left( \right)\left( \right)\ & = \left( \right)\left( \right)\ & = \left( 3 \right)\left( \right)\ & = \frac\end\] This limit almost looks the same as that in the fact in the sense that the argument of the sine is the same as what is in the denominator.
This is not the problem it appears to be once we notice that, \[\frac = \frac\] and then all we need to do is recall a nice property of limits that allows us to do , \[\begin\mathop \limits_ \frac & = \mathop \limits_ \frac\\ & = \frac\\ & = \frac\end\] With a little rewriting we can see that we do in fact end up needing to do a limit like the one we did in the previous part.
So, let’s do the limit here and this time we won’t bother with a change of variable to help us out.
However, with a change of variables we can see that this limit is in fact set to use the fact above regardless.
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.
Note that rules (3) to (6) can be proven using the quotient rule along with the given function expressed in terms of the sine and cosine functions, as illustrated in the following example.
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With this section we’re going to start looking at the derivatives of functions other than polynomials or roots of polynomials.