Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (10+x^2-7*x)/(8-x^3)
-
Limit of (-3-5*x+2*x^2)/(-9+x^2)
-
Limit of (1+x^2)^(2/x)
-
Limit of (-9-x)/(-2+x)
-
Identical expressions
- (one +x^ two)^(two /x)
- (1 plus x squared ) to the power of (2 divide by x)
- (one plus x to the power of two) to the power of (two divide by x)
- (1+x2)(2/x)
- 1+x22/x
- (1+x²)^(2/x)
- (1+x to the power of 2) to the power of (2/x)
- 1+x^2^2/x
- (1+x^2)^(2 divide by x)
-
Similar expressions
- (1-x^2)^(2/x)
Limit of the function (1+x^2)^(2/x)
The solution
You have entered
[src]
2
-
x
/ 2\
lim \1 + x /
x->0+
$$\lim_{x \to 0^+} \left(x^{2} + 1\right)^{\frac{2}{x}}$$
Limit((1 + x^2)^(2/x), x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+} \left(x^{2} + 1\right)^{\frac{2}{x}}$$
transform
do replacement
$$u = \frac{1}{x^{2}}$$
then
$$\lim_{x \to 0^+} \left(1 + \frac{1}{\frac{1}{x^{2}}}\right)^{\frac{2}{x}}$$ =
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{- \frac{2}{\sqrt{\frac{1}{u}}}}$$
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{- \frac{2}{\sqrt{\frac{1}{u}}}}$$
=
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{- \frac{2}{u \sqrt{\frac{1}{u}}}}$$
The limit
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{- \frac{2}{u \sqrt{\frac{1}{u}}}} = e^{- \frac{2}{u \sqrt{\frac{1}{u}}}}$$
The final answer:
$$\lim_{x \to 0^+} \left(x^{2} + 1\right)^{\frac{2}{x}} = 1$$
$$\lim_{x \to 0^+} \left(x^{2} + 1\right)^{\frac{2}{x}}$$
transform
do replacement
$$u = \frac{1}{x^{2}}$$
then
$$\lim_{x \to 0^+} \left(1 + \frac{1}{\frac{1}{x^{2}}}\right)^{\frac{2}{x}}$$ =
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{- \frac{2}{\sqrt{\frac{1}{u}}}}$$
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{- \frac{2}{\sqrt{\frac{1}{u}}}}$$
=
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{- \frac{2}{u \sqrt{\frac{1}{u}}}}$$
The limit
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{- \frac{2}{u \sqrt{\frac{1}{u}}}} = e^{- \frac{2}{u \sqrt{\frac{1}{u}}}}$$
The final answer:
$$\lim_{x \to 0^+} \left(x^{2} + 1\right)^{\frac{2}{x}} = 1$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-} \left(x^{2} + 1\right)^{\frac{2}{x}} = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(x^{2} + 1\right)^{\frac{2}{x}} = 1$$
$$\lim_{x \to \infty} \left(x^{2} + 1\right)^{\frac{2}{x}} = 1$$
More at x→oo
$$\lim_{x \to 1^-} \left(x^{2} + 1\right)^{\frac{2}{x}} = 4$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(x^{2} + 1\right)^{\frac{2}{x}} = 4$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(x^{2} + 1\right)^{\frac{2}{x}} = 1$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+} \left(x^{2} + 1\right)^{\frac{2}{x}} = 1$$
$$\lim_{x \to \infty} \left(x^{2} + 1\right)^{\frac{2}{x}} = 1$$
More at x→oo
$$\lim_{x \to 1^-} \left(x^{2} + 1\right)^{\frac{2}{x}} = 4$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(x^{2} + 1\right)^{\frac{2}{x}} = 4$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(x^{2} + 1\right)^{\frac{2}{x}} = 1$$
More at x→-oo
One‐sided limits
[src]
2
-
x
/ 2\
lim \1 + x /
x->0+
$$\lim_{x \to 0^+} \left(x^{2} + 1\right)^{\frac{2}{x}}$$
1
$$1$$
= 1.0
2
-
x
/ 2\
lim \1 + x /
x->0-
$$\lim_{x \to 0^-} \left(x^{2} + 1\right)^{\frac{2}{x}}$$
1
$$1$$
= 1.0
= 1.0
The graph