Mister Exam
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How to use it?
- Limit of the function:
-
Limit of (5+3*x)/(-5+x)
-
Limit of (-9+x^2)/(15+x^2-8*x)
-
Limit of (-1+x)/log(x)
- Limit of (a+x^2-x*(1+a))/(x^3-a^3)
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Identical expressions
- (- one +x)/log(x)
- ( minus 1 plus x) divide by logarithm of (x)
- ( minus one plus x) divide by logarithm of (x)
- -1+x/logx
- (-1+x) divide by log(x)
-
Similar expressions
- sin(-1+e^(-1+x))/log(x)
- x/(-1+x*sin(-1+e^(-1+x))/log(x))
- (1+x)/log(x)
- (x*log(x)-(-1+x)*log(-1+x))/log(x)
- (-1-x)/log(x)
Limit of the function (-1+x)/log(x)
The solution
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[src]
/-1 + x\ lim |------| x->1+\log(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
i.e. limit for the numerator is
$$\lim_{x \to 1^+}\left(x - 1\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 1^+} \log{\left(x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 1^+}\left(\frac{x - 1}{\log{\left(x \right)}}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(x - 1\right)}{\frac{d}{d x} \log{\left(x \right)}}\right)$$
=
$$\lim_{x \to 1^+} x$$
=
$$\lim_{x \to 1^+} x$$
=
$$1$$
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)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 1^+}\left(x - 1\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 1^+} \log{\left(x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 1^+}\left(\frac{x - 1}{\log{\left(x \right)}}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(x - 1\right)}{\frac{d}{d x} \log{\left(x \right)}}\right)$$
=
$$\lim_{x \to 1^+} x$$
=
$$\lim_{x \to 1^+} x$$
=
$$1$$
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
One‐sided limits
[src]
/-1 + x\ lim |------| x->1+\log(x)/
$$\lim_{x \to 1^+}\left(\frac{x - 1}{\log{\left(x \right)}}\right)$$
1
$$1$$
= 1.0
/-1 + x\ lim |------| x->1-\log(x)/
$$\lim_{x \to 1^-}\left(\frac{x - 1}{\log{\left(x \right)}}\right)$$
1
$$1$$
= 1.0
= 1.0
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 1^-}\left(\frac{x - 1}{\log{\left(x \right)}}\right) = 1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x - 1}{\log{\left(x \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{x - 1}{\log{\left(x \right)}}\right) = \infty$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{x - 1}{\log{\left(x \right)}}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x - 1}{\log{\left(x \right)}}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to -\infty}\left(\frac{x - 1}{\log{\left(x \right)}}\right) = -\infty$$
More at x→-oo
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x - 1}{\log{\left(x \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{x - 1}{\log{\left(x \right)}}\right) = \infty$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{x - 1}{\log{\left(x \right)}}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x - 1}{\log{\left(x \right)}}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to -\infty}\left(\frac{x - 1}{\log{\left(x \right)}}\right) = -\infty$$
More at x→-oo
The graph