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(-1+x)/log(x)
Limit of the function/ (-1+x)/log(x)

Limit of the function (-1+x)/log(x)

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

i.e. limit for the numerator is
lim⁡x→1+(x−1)=0\lim_{x \to 1^+}\left(x - 1\right) = 0
and limit for the denominator is
lim⁡x→1+log⁡(x)=0\lim_{x \to 1^+} \log{\left(x \right)} = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→1+(x−1log⁡(x))\lim_{x \to 1^+}\left(\frac{x - 1}{\log{\left(x \right)}}\right)
=
lim⁡x→1+(ddx(x−1)ddxlog⁡(x))\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(x - 1\right)}{\frac{d}{d x} \log{\left(x \right)}}\right)
=
lim⁡x→1+x\lim_{x \to 1^+} x
=
lim⁡x→1+x\lim_{x \to 1^+} x
=
11
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
-2.0-1.5-1.0-0.52.00.00.51.01.502
Plot the graph
One‐sided limits [src] 
     /-1 + x\
 lim |------|
x->1+\log(x)/
lim⁡x→1+(x−1log⁡(x))\lim_{x \to 1^+}\left(\frac{x - 1}{\log{\left(x \right)}}\right) 
1
11 
= 1.0
     /-1 + x\
 lim |------|
x->1-\log(x)/
lim⁡x→1−(x−1log⁡(x))\lim_{x \to 1^-}\left(\frac{x - 1}{\log{\left(x \right)}}\right) 
1
11 
= 1.0
= 1.0
Rapid solution [src] 
1
11
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→1−(x−1log⁡(x))=1\lim_{x \to 1^-}\left(\frac{x - 1}{\log{\left(x \right)}}\right) = 1
More at x→1 from the left
lim⁡x→1+(x−1log⁡(x))=1\lim_{x \to 1^+}\left(\frac{x - 1}{\log{\left(x \right)}}\right) = 1
lim⁡x→∞(x−1log⁡(x))=∞\lim_{x \to \infty}\left(\frac{x - 1}{\log{\left(x \right)}}\right) = \infty
More at x→oo
lim⁡x→0−(x−1log⁡(x))=0\lim_{x \to 0^-}\left(\frac{x - 1}{\log{\left(x \right)}}\right) = 0
More at x→0 from the left
lim⁡x→0+(x−1log⁡(x))=0\lim_{x \to 0^+}\left(\frac{x - 1}{\log{\left(x \right)}}\right) = 0
More at x→0 from the right
lim⁡x→−∞(x−1log⁡(x))=−∞\lim_{x \to -\infty}\left(\frac{x - 1}{\log{\left(x \right)}}\right) = -\infty
More at x→-oo
Numerical answer [src] 
1.0
1.0
The graph
Limit of the function (-1+x)/log(x)
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