Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (-5+7*n)/(2+n)
-
Limit of (-64+4^x)/(-3+x)
-
Limit of (-7+3*x^2+5*x)/(1+x+3*x^2)
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Limit of (-2+2*x^3+7*x)/(-4-x+3*x^3)
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Identical expressions
- three ^(-x)*log(x)
- 3 to the power of ( minus x) multiply by logarithm of (x)
- three to the power of ( minus x) multiply by logarithm of (x)
- 3(-x)*log(x)
- 3-x*logx
- 3^(-x)log(x)
- 3(-x)log(x)
- 3-xlogx
- 3^-xlogx
-
Similar expressions
- 3^(x)*log(x)
Limit of the function 3^(-x)*log(x)
The solution
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/ -x \ lim \3 *log(x)/ x->oo
$$\lim_{x \to \infty}\left(3^{- x} \log{\left(x \right)}\right)$$
Limit(3^(-x)*log(x), x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \infty} \log{\left(x \right)} = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty} 3^{x} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(3^{- x} \log{\left(x \right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \log{\left(x \right)}}{\frac{d}{d x} 3^{x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{3^{- x}}{x \log{\left(3 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{3^{- x}}{x \log{\left(3 \right)}}\right)$$
=
$$0$$
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)
oo/oo,
i.e. limit for the numerator is
$$\lim_{x \to \infty} \log{\left(x \right)} = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty} 3^{x} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(3^{- x} \log{\left(x \right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \log{\left(x \right)}}{\frac{d}{d x} 3^{x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{3^{- x}}{x \log{\left(3 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{3^{- x}}{x \log{\left(3 \right)}}\right)$$
=
$$0$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(3^{- x} \log{\left(x \right)}\right) = 0$$
$$\lim_{x \to 0^-}\left(3^{- x} \log{\left(x \right)}\right) = -\infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(3^{- x} \log{\left(x \right)}\right) = -\infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(3^{- x} \log{\left(x \right)}\right) = 0$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(3^{- x} \log{\left(x \right)}\right) = 0$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(3^{- x} \log{\left(x \right)}\right) = \infty$$
More at x→-oo
$$\lim_{x \to 0^-}\left(3^{- x} \log{\left(x \right)}\right) = -\infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(3^{- x} \log{\left(x \right)}\right) = -\infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(3^{- x} \log{\left(x \right)}\right) = 0$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(3^{- x} \log{\left(x \right)}\right) = 0$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(3^{- x} \log{\left(x \right)}\right) = \infty$$
More at x→-oo
The graph