Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (4-x^2)/(3-x^2)
-
Limit of (-2+x^2-x)/(-2+x+3*x^2)
-
Limit of 5-9*x+3*x^2/2
-
Limit of (-16+x^2)/(-64+x^3)
- Sum of series:
-
3^n/n^2
-
Identical expressions
- three ^n/n^ two
- 3 to the power of n divide by n squared
- three to the power of n divide by n to the power of two
- 3n/n2
- 3^n/n²
- 3 to the power of n/n to the power of 2
- 3^n divide by n^2
Limit of the function 3^n/n^2
The solution
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/ n\
|3 |
lim |--|
n->oo| 2|
\n /
$$\lim_{n \to \infty}\left(\frac{3^{n}}{n^{2}}\right)$$
Limit(3^n/n^2, n, oo, dir='-')
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{n \to \infty} 3^{n} = \infty$$
and limit for the denominator is
$$\lim_{n \to \infty} n^{2} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{n \to \infty}\left(\frac{3^{n}}{n^{2}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} 3^{n}}{\frac{d}{d n} n^{2}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{3^{n} \log{\left(3 \right)}}{2 n}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{3^{n} \log{\left(3 \right)}}{2}}{\frac{d}{d n} n}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{3^{n} \log{\left(3 \right)}^{2}}{2}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \log{\left(3 \right)}^{2}}{\frac{d}{d n} 2 \cdot 3^{- n}}\right)$$
=
$$\lim_{n \to \infty} 0$$
=
$$\lim_{n \to \infty} 0$$
=
$$\infty$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 3 time(s)
oo/oo,
i.e. limit for the numerator is
$$\lim_{n \to \infty} 3^{n} = \infty$$
and limit for the denominator is
$$\lim_{n \to \infty} n^{2} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{n \to \infty}\left(\frac{3^{n}}{n^{2}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} 3^{n}}{\frac{d}{d n} n^{2}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{3^{n} \log{\left(3 \right)}}{2 n}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{3^{n} \log{\left(3 \right)}}{2}}{\frac{d}{d n} n}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{3^{n} \log{\left(3 \right)}^{2}}{2}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \log{\left(3 \right)}^{2}}{\frac{d}{d n} 2 \cdot 3^{- n}}\right)$$
=
$$\lim_{n \to \infty} 0$$
=
$$\lim_{n \to \infty} 0$$
=
$$\infty$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 3 time(s)
The graph
Other limits n→0, -oo, +oo, 1
$$\lim_{n \to \infty}\left(\frac{3^{n}}{n^{2}}\right) = \infty$$
$$\lim_{n \to 0^-}\left(\frac{3^{n}}{n^{2}}\right) = \infty$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(\frac{3^{n}}{n^{2}}\right) = \infty$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(\frac{3^{n}}{n^{2}}\right) = 3$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(\frac{3^{n}}{n^{2}}\right) = 3$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(\frac{3^{n}}{n^{2}}\right) = 0$$
More at n→-oo
$$\lim_{n \to 0^-}\left(\frac{3^{n}}{n^{2}}\right) = \infty$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(\frac{3^{n}}{n^{2}}\right) = \infty$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(\frac{3^{n}}{n^{2}}\right) = 3$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(\frac{3^{n}}{n^{2}}\right) = 3$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(\frac{3^{n}}{n^{2}}\right) = 0$$
More at n→-oo
The graph