Mister Exam

Other calculators:

log(cos(x))/x^2
Limit of the function/ log(cos(x))/x^2

Limit of the function log(cos(x))/x^2

at →
v

For end points:

The graph:

from to

Piecewise:

The solution

You have entered [src] 
     /log(cos(x))\
 lim |-----------|
x->0+|      2    |
     \     x     /
lim⁡x→0+(log⁡(cos⁡(x))x2)\lim_{x \to 0^+}\left(\frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) 
Limit(log(cos(x))/x^2, x, 0)
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→0+log⁡(cos⁡(x))=0\lim_{x \to 0^+} \log{\left(\cos{\left(x \right)} \right)} = 0
and limit for the denominator is
lim⁡x→0+x2=0\lim_{x \to 0^+} x^{2} = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→0+(log⁡(cos⁡(x))x2)\lim_{x \to 0^+}\left(\frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right)
=
lim⁡x→0+(ddxlog⁡(cos⁡(x))ddxx2)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \log{\left(\cos{\left(x \right)} \right)}}{\frac{d}{d x} x^{2}}\right)
=
lim⁡x→0+(−sin⁡(x)2xcos⁡(x))\lim_{x \to 0^+}\left(- \frac{\sin{\left(x \right)}}{2 x \cos{\left(x \right)}}\right)
=
lim⁡x→0+(−sin⁡(x)2x)\lim_{x \to 0^+}\left(- \frac{\sin{\left(x \right)}}{2 x}\right)
=
lim⁡x→0+(−sin⁡(x)2x)\lim_{x \to 0^+}\left(- \frac{\sin{\left(x \right)}}{2 x}\right)
=
−12- \frac{1}{2}
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
02468-8-6-4-2-10101-2
Plot the graph
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(log⁡(cos⁡(x))x2)=−12\lim_{x \to 0^-}\left(\frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) = - \frac{1}{2}
More at x→0 from the left
lim⁡x→0+(log⁡(cos⁡(x))x2)=−12\lim_{x \to 0^+}\left(\frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) = - \frac{1}{2}
lim⁡x→∞(log⁡(cos⁡(x))x2)=0\lim_{x \to \infty}\left(\frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) = 0
More at x→oo
lim⁡x→1−(log⁡(cos⁡(x))x2)=log⁡(cos⁡(1))\lim_{x \to 1^-}\left(\frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) = \log{\left(\cos{\left(1 \right)} \right)}
More at x→1 from the left
lim⁡x→1+(log⁡(cos⁡(x))x2)=log⁡(cos⁡(1))\lim_{x \to 1^+}\left(\frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) = \log{\left(\cos{\left(1 \right)} \right)}
More at x→1 from the right
lim⁡x→−∞(log⁡(cos⁡(x))x2)=0\lim_{x \to -\infty}\left(\frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) = 0
More at x→-oo
Rapid solution [src] 
-1/2
−12- \frac{1}{2}
Expand and simplify
One‐sided limits [src] 
     /log(cos(x))\
 lim |-----------|
x->0+|      2    |
     \     x     /
lim⁡x→0+(log⁡(cos⁡(x))x2)\lim_{x \to 0^+}\left(\frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) 
-1/2
−12- \frac{1}{2} 
= -0.5
     /log(cos(x))\
 lim |-----------|
x->0-|      2    |
     \     x     /
lim⁡x→0−(log⁡(cos⁡(x))x2)\lim_{x \to 0^-}\left(\frac{\log{\left(\cos{\left(x \right)} \right)}}{x^{2}}\right) 
-1/2
−12- \frac{1}{2} 
= -0.5
= -0.5
Numerical answer [src] 
-0.5
-0.5
The graph
Limit of the function log(cos(x))/x^2
    © Mister Exam – Calculator