Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of 10+x^2+3*x^3+8*x
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Limit of (-16+2^x)/(-1+5*sqrt(x)*(5-x))
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Limit of (-4+x^2)/(-3+sqrt(1-4*x))
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Limit of ((1+x)/(-2+x))^(3+x)
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Identical expressions
- (- one +(one +x)^a)/x
- ( minus 1 plus (1 plus x) to the power of a) divide by x
- ( minus one plus (one plus x) to the power of a) divide by x
- (-1+(1+x)a)/x
- -1+1+xa/x
- -1+1+x^a/x
- (-1+(1+x)^a) divide by x
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Similar expressions
- (1+(1+x)^a)/x
- (-1+(1-x)^a)/x
- (-1-(1+x)^a)/x
Limit of the function (-1+(1+x)^a)/x
The solution
You have entered
[src]
/ a\
|-1 + (1 + x) |
lim |-------------|
x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{\left(x + 1\right)^{a} - 1}{x}\right)$$
Limit((-1 + (1 + x)^a)/x, x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(\left(x + 1\right)^{a} - 1\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\left(x + 1\right)^{a} - 1}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(x + 1\right)^{a} - 1}{x}\right)$$
=
$$a$$
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
$$\lim_{x \to 0^+}\left(\left(x + 1\right)^{a} - 1\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\left(x + 1\right)^{a} - 1}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(x + 1\right)^{a} - 1}{x}\right)$$
=
$$a$$
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]
/ a\
|-1 + (1 + x) |
lim |-------------|
x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{\left(x + 1\right)^{a} - 1}{x}\right)$$
a
$$a$$
/ a\
|-1 + (1 + x) |
lim |-------------|
x->0-\ x /
$$\lim_{x \to 0^-}\left(\frac{\left(x + 1\right)^{a} - 1}{x}\right)$$
a
$$a$$
a
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\left(x + 1\right)^{a} - 1}{x}\right) = a$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\left(x + 1\right)^{a} - 1}{x}\right) = a$$
$$\lim_{x \to \infty}\left(\frac{\left(x + 1\right)^{a} - 1}{x}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\left(x + 1\right)^{a} - 1}{x}\right) = 2^{a} - 1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\left(x + 1\right)^{a} - 1}{x}\right) = 2^{a} - 1$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\left(x + 1\right)^{a} - 1}{x}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\left(x + 1\right)^{a} - 1}{x}\right) = a$$
$$\lim_{x \to \infty}\left(\frac{\left(x + 1\right)^{a} - 1}{x}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\left(x + 1\right)^{a} - 1}{x}\right) = 2^{a} - 1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\left(x + 1\right)^{a} - 1}{x}\right) = 2^{a} - 1$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\left(x + 1\right)^{a} - 1}{x}\right)$$
More at x→-oo