Evaluate the limit of (sin 4x + cos 2x)/(4x - pi) as x approaches pi/4.

Step-by-step solution

1. 1

   Check the form. At $x=\frac{\pi}{4}$ we get $\sin\pi+\cos\frac{\pi}{2}=0+0=0$ on top and $\pi-\pi=0$ underneath, so the form is $\frac{0}{0}$.

2. 2

   Shift the variable to the origin. Put $x=\frac{\pi}{4}+t$ with $t\to 0$. The denominator becomes $4t$, which is the simplest possible form to divide by.

3. 3

   Rewrite the numerator. $\sin 4x=\sin(\pi+4t)=-\sin 4t$ and $\cos 2x=\cos\left(\frac{\pi}{2}+2t\right)=-\sin 2t$, so the numerator is $-\sin 4t-\sin 2t$.

4. 4

   Divide by $t$. The quotient equals $\frac{-\frac{\sin 4t}{t}-\frac{\sin 2t}{t}}{4}$, and by the standard limit $\frac{\sin kt}{t}\to k$ the numerator tends to $-4-2=-6$.

5. 5

   Finish. The limit is $\frac{-6}{4}=-\frac{3}{2}$. Differentiating top and bottom once (L'Hopital) gives $\frac{4\cos 4x-2\sin 2x}{4}\to\frac{-4-2}{4}$, the same value.

6. 6

   Numerical check. At $x=\frac{\pi}{4}+10^{-7}$ the original expression evaluates to $-1.5000000$, confirming $-\frac{3}{2}$.