Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (x+sqrt(10+x-x^2))/(sqrt(2-7*x)+2*x)
- Limit of (-1+x^m)/(-1+x)
- Limit of x*a^(-x)
-
Limit of -4+x+x^3
-
Identical expressions
- (- one +x^m)/(- one +x)
- ( minus 1 plus x to the power of m) divide by ( minus 1 plus x)
- ( minus one plus x to the power of m) divide by ( minus one plus x)
- (-1+xm)/(-1+x)
- -1+xm/-1+x
- -1+x^m/-1+x
- (-1+x^m) divide by (-1+x)
-
Similar expressions
- (-1+x^m)/(-1-x)
- (-1-x^m)/(-1+x)
- (-1+x^m)/(1+x)
- (1+x^m)/(-1+x)
Limit of the function (-1+x^m)/(-1+x)
The solution
You have entered
[src]
/ m\
|-1 + x |
lim |-------|
x->1+\ -1 + x/
$$\lim_{x \to 1^+}\left(\frac{x^{m} - 1}{x - 1}\right)$$
Limit((-1 + x^m)/(-1 + x), x, 1)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 1^+}\left(x^{m} - 1\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 1^+}\left(x - 1\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 1^+}\left(\frac{x^{m} - 1}{x - 1}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{x^{m} - 1}{x - 1}\right)$$
=
$$m$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 0 time(s)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 1^+}\left(x^{m} - 1\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 1^+}\left(x - 1\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 1^+}\left(\frac{x^{m} - 1}{x - 1}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{x^{m} - 1}{x - 1}\right)$$
=
$$m$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 0 time(s)
One‐sided limits
[src]
/ m\
|-1 + x |
lim |-------|
x->1+\ -1 + x/
$$\lim_{x \to 1^+}\left(\frac{x^{m} - 1}{x - 1}\right)$$
m
$$m$$
/ m\
|-1 + x |
lim |-------|
x->1-\ -1 + x/
$$\lim_{x \to 1^-}\left(\frac{x^{m} - 1}{x - 1}\right)$$
m
$$m$$
m
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 1^-}\left(\frac{x^{m} - 1}{x - 1}\right) = m$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x^{m} - 1}{x - 1}\right) = m$$
$$\lim_{x \to \infty}\left(\frac{x^{m} - 1}{x - 1}\right)$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{x^{m} - 1}{x - 1}\right)$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x^{m} - 1}{x - 1}\right)$$
More at x→0 from the right
$$\lim_{x \to -\infty}\left(\frac{x^{m} - 1}{x - 1}\right)$$
More at x→-oo
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x^{m} - 1}{x - 1}\right) = m$$
$$\lim_{x \to \infty}\left(\frac{x^{m} - 1}{x - 1}\right)$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{x^{m} - 1}{x - 1}\right)$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x^{m} - 1}{x - 1}\right)$$
More at x→0 from the right
$$\lim_{x \to -\infty}\left(\frac{x^{m} - 1}{x - 1}\right)$$
More at x→-oo