Mister Exam

Other calculators:

x/(x-4*x^2)
Limit of the function/ x/(x-4*x^2)

Limit of the function x/(x-4*x^2)

at →
v

For end points:

The graph:

from to

Piecewise:

The solution

You have entered [src] 
     /   x    \
 lim |--------|
x->0+|       2|
     \x - 4*x /
lim⁡x→0+(x−4x2+x)\lim_{x \to 0^+}\left(\frac{x}{- 4 x^{2} + x}\right) 
Limit(x/(x - 4*x^2), x, 0)
Detail solution
Let's take the limit
lim⁡x→0+(x−4x2+x)\lim_{x \to 0^+}\left(\frac{x}{- 4 x^{2} + x}\right)
transform
lim⁡x→0+(x−4x2+x)\lim_{x \to 0^+}\left(\frac{x}{- 4 x^{2} + x}\right)
=
lim⁡x→0+(x(−1)x(4x−1))\lim_{x \to 0^+}\left(\frac{x}{\left(-1\right) x \left(4 x - 1\right)}\right)
=
lim⁡x→0+(−14x−1)=\lim_{x \to 0^+}\left(- \frac{1}{4 x - 1}\right) =
−1−1+4⋅0=- \frac{1}{-1 + 4 \cdot 0} = 
= 1

The final answer:
lim⁡x→0+(x−4x2+x)=1\lim_{x \to 0^+}\left(\frac{x}{- 4 x^{2} + x}\right) = 1
Lopital's rule
There is no sense to apply Lopital's rule to this function since there is no indeterminateness of 0/0 or oo/oo type
The graph
02468-8-6-4-2-10105-5
Plot the graph
Rapid solution [src] 
1
11
Expand and simplify
One‐sided limits [src] 
     /   x    \
 lim |--------|
x->0+|       2|
     \x - 4*x /
lim⁡x→0+(x−4x2+x)\lim_{x \to 0^+}\left(\frac{x}{- 4 x^{2} + x}\right) 
1
11 
     /   x    \
 lim |--------|
x->0-|       2|
     \x - 4*x /
lim⁡x→0−(x−4x2+x)\lim_{x \to 0^-}\left(\frac{x}{- 4 x^{2} + x}\right) 
1
11 
= 1
= 1
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(x−4x2+x)=1\lim_{x \to 0^-}\left(\frac{x}{- 4 x^{2} + x}\right) = 1
More at x→0 from the left
lim⁡x→0+(x−4x2+x)=1\lim_{x \to 0^+}\left(\frac{x}{- 4 x^{2} + x}\right) = 1
lim⁡x→∞(x−4x2+x)=0\lim_{x \to \infty}\left(\frac{x}{- 4 x^{2} + x}\right) = 0
More at x→oo
lim⁡x→1−(x−4x2+x)=−13\lim_{x \to 1^-}\left(\frac{x}{- 4 x^{2} + x}\right) = - \frac{1}{3}
More at x→1 from the left
lim⁡x→1+(x−4x2+x)=−13\lim_{x \to 1^+}\left(\frac{x}{- 4 x^{2} + x}\right) = - \frac{1}{3}
More at x→1 from the right
lim⁡x→−∞(x−4x2+x)=0\lim_{x \to -\infty}\left(\frac{x}{- 4 x^{2} + x}\right) = 0
More at x→-oo
The graph
Limit of the function x/(x-4*x^2)
    © Mister Exam – Calculator