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-2+|-2+x|/x
Limit of the function/ -2+|-2+x|/x

Limit of the function -2+|-2+x|/x

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     /     |-2 + x|\
 lim |-2 + --------|
x->2+\        x    /
lim⁡x→2+(−2+∣x−2∣x)\lim_{x \to 2^+}\left(-2 + \frac{\left|{x - 2}\right|}{x}\right) 
Limit(-2 + |-2 + x|/x, x, 2)
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
-4.0-3.0-2.0-1.04.00.01.02.03.0-100100
Plot the graph
Rapid solution [src] 
-2
−2-2
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→2−(−2+∣x−2∣x)=−2\lim_{x \to 2^-}\left(-2 + \frac{\left|{x - 2}\right|}{x}\right) = -2
More at x→2 from the left
lim⁡x→2+(−2+∣x−2∣x)=−2\lim_{x \to 2^+}\left(-2 + \frac{\left|{x - 2}\right|}{x}\right) = -2
lim⁡x→∞(−2+∣x−2∣x)=−1\lim_{x \to \infty}\left(-2 + \frac{\left|{x - 2}\right|}{x}\right) = -1
More at x→oo
lim⁡x→0−(−2+∣x−2∣x)=−∞\lim_{x \to 0^-}\left(-2 + \frac{\left|{x - 2}\right|}{x}\right) = -\infty
More at x→0 from the left
lim⁡x→0+(−2+∣x−2∣x)=∞\lim_{x \to 0^+}\left(-2 + \frac{\left|{x - 2}\right|}{x}\right) = \infty
More at x→0 from the right
lim⁡x→1−(−2+∣x−2∣x)=−1\lim_{x \to 1^-}\left(-2 + \frac{\left|{x - 2}\right|}{x}\right) = -1
More at x→1 from the left
lim⁡x→1+(−2+∣x−2∣x)=−1\lim_{x \to 1^+}\left(-2 + \frac{\left|{x - 2}\right|}{x}\right) = -1
More at x→1 from the right
lim⁡x→−∞(−2+∣x−2∣x)=−3\lim_{x \to -\infty}\left(-2 + \frac{\left|{x - 2}\right|}{x}\right) = -3
More at x→-oo
One‐sided limits [src] 
     /     |-2 + x|\
 lim |-2 + --------|
x->2+\        x    /
lim⁡x→2+(−2+∣x−2∣x)\lim_{x \to 2^+}\left(-2 + \frac{\left|{x - 2}\right|}{x}\right) 
-2
−2-2 
= -2.0
     /     |-2 + x|\
 lim |-2 + --------|
x->2-\        x    /
lim⁡x→2−(−2+∣x−2∣x)\lim_{x \to 2^-}\left(-2 + \frac{\left|{x - 2}\right|}{x}\right) 
-2
−2-2 
= -2.0
= -2.0
Numerical answer [src] 
-2.0
-2.0
The graph
Limit of the function -2+|-2+x|/x
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