1 Answer

Matousc · Answer 1 · 2023-04-17T23:38:52+0000

1) If the limit
is

$L = \displaystyle \lim_(\Delta x\to0) (\sin\left(\frac\pi3 + \Delta x\right) - \sin\left(\frac\pi3\right))/(\Delta x)$

then using the hint as well as
$\sin\left(\frac\pi3\right)=\frac{\sqrt3}2$ and
$\cos\left(\frac\pi3\right)=\frac12$ we have

$\displaystyle L = \lim_(\Delta x\to0) (\sin\left(\frac\pi3\right)\cos(\Delta x) + \cos\left(\frac\pi3\right)\sin(\Delta x) - \sin\left(\frac\pi3\right))/(\Delta x)$

$\displaystyle L = \sin\left(\frac\pi3\right) \lim_(\Delta x\to0) (\cos(\Delta x) - 1)/(\Delta x) + \cos\left(\frac\pi3\right) \lim_(\Delta x) (\sin(\Delta x))/(\Delta x)$

$L = \frac{\sqrt3}2 * 0 + \frac12*1 = \boxed{\frac12}$

which follows from the well-known limits,

$\displaystyle \lim_(x\to0) \frac{1-\cos(x)}x = 0 \text{ and } \lim_(x\to0) \frac{\sin(x)}x=1$

Alternatively, if you already know about derivatives, we can identify the limit as the derivative of
$\sin(x)$ at
$x=\frac\pi3$ , which is
$\cos\left(\frac\pi3\right)=\frac12$ .

2) It looks like you may be using double square brackets deliberately to denote the greatest integer or floor function which rounds the input down to the nearest integer. That is,
$[\![x]\!]$ is the greatest integer that is less than or equal to
. The existence of
depends on the equality of the one-sided limits.