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(1-sin(2*x))/(pi-4*x)
Limit of the function/ (1-sin(2*x))/(pi-4*x)

Limit of the function (1-sin(2*x))/(pi-4*x)

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The solution

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      /1 - sin(2*x)\
 lim  |------------|
   pi \  pi - 4*x  /
x->--+              
   4
lim⁡x→π4+(1−sin⁡(2x)π−4x)\lim_{x \to \frac{\pi}{4}^+}\left(\frac{1 - \sin{\left(2 x \right)}}{\pi - 4 x}\right) 
Limit((1 - sin(2*x))/(pi - 4*x), x, pi/4)
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→π4+(1−sin⁡(2x))=0\lim_{x \to \frac{\pi}{4}^+}\left(1 - \sin{\left(2 x \right)}\right) = 0
and limit for the denominator is
lim⁡x→π4+(π−4x)=0\lim_{x \to \frac{\pi}{4}^+}\left(\pi - 4 x\right) = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→π4+(1−sin⁡(2x)π−4x)\lim_{x \to \frac{\pi}{4}^+}\left(\frac{1 - \sin{\left(2 x \right)}}{\pi - 4 x}\right)
=
lim⁡x→π4+(ddx(1−sin⁡(2x))ddx(π−4x))\lim_{x \to \frac{\pi}{4}^+}\left(\frac{\frac{d}{d x} \left(1 - \sin{\left(2 x \right)}\right)}{\frac{d}{d x} \left(\pi - 4 x\right)}\right)
=
lim⁡x→π4+(cos⁡(2x)2)\lim_{x \to \frac{\pi}{4}^+}\left(\frac{\cos{\left(2 x \right)}}{2}\right)
=
lim⁡x→π4+(cos⁡(2x)2)\lim_{x \to \frac{\pi}{4}^+}\left(\frac{\cos{\left(2 x \right)}}{2}\right)
=
00
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)
The graph
-1.50-1.25-1.00-0.75-0.50-0.250.000.250.500.751.001.251.500.5-0.5
Plot the graph
Rapid solution [src] 
0
00
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→π4−(1−sin⁡(2x)π−4x)=0\lim_{x \to \frac{\pi}{4}^-}\left(\frac{1 - \sin{\left(2 x \right)}}{\pi - 4 x}\right) = 0
More at x→pi/4 from the left
lim⁡x→π4+(1−sin⁡(2x)π−4x)=0\lim_{x \to \frac{\pi}{4}^+}\left(\frac{1 - \sin{\left(2 x \right)}}{\pi - 4 x}\right) = 0
lim⁡x→∞(1−sin⁡(2x)π−4x)=0\lim_{x \to \infty}\left(\frac{1 - \sin{\left(2 x \right)}}{\pi - 4 x}\right) = 0
More at x→oo
lim⁡x→0−(1−sin⁡(2x)π−4x)=1π\lim_{x \to 0^-}\left(\frac{1 - \sin{\left(2 x \right)}}{\pi - 4 x}\right) = \frac{1}{\pi}
More at x→0 from the left
lim⁡x→0+(1−sin⁡(2x)π−4x)=1π\lim_{x \to 0^+}\left(\frac{1 - \sin{\left(2 x \right)}}{\pi - 4 x}\right) = \frac{1}{\pi}
More at x→0 from the right
lim⁡x→1−(1−sin⁡(2x)π−4x)=−−1+sin⁡(2)−4+π\lim_{x \to 1^-}\left(\frac{1 - \sin{\left(2 x \right)}}{\pi - 4 x}\right) = - \frac{-1 + \sin{\left(2 \right)}}{-4 + \pi}
More at x→1 from the left
lim⁡x→1+(1−sin⁡(2x)π−4x)=−−1+sin⁡(2)−4+π\lim_{x \to 1^+}\left(\frac{1 - \sin{\left(2 x \right)}}{\pi - 4 x}\right) = - \frac{-1 + \sin{\left(2 \right)}}{-4 + \pi}
More at x→1 from the right
lim⁡x→−∞(1−sin⁡(2x)π−4x)=0\lim_{x \to -\infty}\left(\frac{1 - \sin{\left(2 x \right)}}{\pi - 4 x}\right) = 0
More at x→-oo
One‐sided limits [src] 
      /1 - sin(2*x)\
 lim  |------------|
   pi \  pi - 4*x  /
x->--+              
   4
lim⁡x→π4+(1−sin⁡(2x)π−4x)\lim_{x \to \frac{\pi}{4}^+}\left(\frac{1 - \sin{\left(2 x \right)}}{\pi - 4 x}\right) 
0
00 
= 1.53080849893419e-17
      /1 - sin(2*x)\
 lim  |------------|
   pi \  pi - 4*x  /
x->---              
   4
lim⁡x→π4−(1−sin⁡(2x)π−4x)\lim_{x \to \frac{\pi}{4}^-}\left(\frac{1 - \sin{\left(2 x \right)}}{\pi - 4 x}\right) 
0
00 
= 1.53080849893419e-17
= 1.53080849893419e-17
Numerical answer [src] 
1.53080849893419e-17
1.53080849893419e-17
The graph
Limit of the function (1-sin(2*x))/(pi-4*x)
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