Mister Exam
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How to use it?
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Limit of (x*(1/2+x))^x
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Limit of (2-sqrt(x))/(4-sqrt(2)*sqrt(x))
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Identical expressions
- (x*(one / two +x))^x
- (x multiply by (1 divide by 2 plus x)) to the power of x
- (x multiply by (one divide by two plus x)) to the power of x
- (x*(1/2+x))x
- x*1/2+xx
- (x(1/2+x))^x
- (x(1/2+x))x
- x1/2+xx
- x1/2+x^x
- (x*(1 divide by 2+x))^x
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Similar expressions
- (x*(1/2-x))^x
Limit of the function (x*(1/2+x))^x
The solution
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x lim (x*(1/2 + x)) x->oo
$$\lim_{x \to \infty} \left(x \left(x + \frac{1}{2}\right)\right)^{x}$$
Limit((x*(1/2 + x))^x, x, oo, dir='-')
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
$$\lim_{x \to \infty} \left(x \left(x + \frac{1}{2}\right)\right)^{x} = \infty$$
$$\lim_{x \to 0^-} \left(x \left(x + \frac{1}{2}\right)\right)^{x} = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(x \left(x + \frac{1}{2}\right)\right)^{x} = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-} \left(x \left(x + \frac{1}{2}\right)\right)^{x} = \frac{3}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(x \left(x + \frac{1}{2}\right)\right)^{x} = \frac{3}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(x \left(x + \frac{1}{2}\right)\right)^{x} = 0$$
More at x→-oo
$$\lim_{x \to 0^-} \left(x \left(x + \frac{1}{2}\right)\right)^{x} = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(x \left(x + \frac{1}{2}\right)\right)^{x} = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-} \left(x \left(x + \frac{1}{2}\right)\right)^{x} = \frac{3}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(x \left(x + \frac{1}{2}\right)\right)^{x} = \frac{3}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(x \left(x + \frac{1}{2}\right)\right)^{x} = 0$$
More at x→-oo
The graph