Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (11+12*x+17*x^3)/(-27-14*x+20*x^3)
-
Limit of 1/(-1+2^(-3+x))
-
Limit of (10+12*x)/x^3
- Limit of (1+2*x/3)^(3251035981008077*x/140737488355328)
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Identical expressions
- six /n^ two + seven /n^ three
- 6 divide by n squared plus 7 divide by n cubed
- six divide by n to the power of two plus seven divide by n to the power of three
- 6/n2+7/n3
- 6/n²+7/n³
- 6/n to the power of 2+7/n to the power of 3
- 6 divide by n^2+7 divide by n^3
-
Similar expressions
- 6/n^2-7/n^3
Limit of the function 6/n^2+7/n^3
The solution
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[src]
/6 7 \
lim |-- + --|
n->oo| 2 3|
\n n /
$$\lim_{n \to \infty}\left(\frac{7}{n^{3}} + \frac{6}{n^{2}}\right)$$
Limit(6/n^2 + 7/n^3, n, oo, dir='-')
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{n \to \infty}\left(6 n + 7\right) = \infty$$
and limit for the denominator is
$$\lim_{n \to \infty} n^{3} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{n \to \infty}\left(\frac{7}{n^{3}} + \frac{6}{n^{2}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{n \to \infty}\left(\frac{6 n + 7}{n^{3}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(6 n + 7\right)}{\frac{d}{d n} n^{3}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{2}{n^{2}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{2}{n^{2}}\right)$$
=
$$0$$
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
$$\lim_{n \to \infty}\left(6 n + 7\right) = \infty$$
and limit for the denominator is
$$\lim_{n \to \infty} n^{3} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{n \to \infty}\left(\frac{7}{n^{3}} + \frac{6}{n^{2}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{n \to \infty}\left(\frac{6 n + 7}{n^{3}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(6 n + 7\right)}{\frac{d}{d n} n^{3}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{2}{n^{2}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{2}{n^{2}}\right)$$
=
$$0$$
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 n→0, -oo, +oo, 1
$$\lim_{n \to \infty}\left(\frac{7}{n^{3}} + \frac{6}{n^{2}}\right) = 0$$
$$\lim_{n \to 0^-}\left(\frac{7}{n^{3}} + \frac{6}{n^{2}}\right) = -\infty$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(\frac{7}{n^{3}} + \frac{6}{n^{2}}\right) = \infty$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(\frac{7}{n^{3}} + \frac{6}{n^{2}}\right) = 13$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(\frac{7}{n^{3}} + \frac{6}{n^{2}}\right) = 13$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(\frac{7}{n^{3}} + \frac{6}{n^{2}}\right) = 0$$
More at n→-oo
$$\lim_{n \to 0^-}\left(\frac{7}{n^{3}} + \frac{6}{n^{2}}\right) = -\infty$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(\frac{7}{n^{3}} + \frac{6}{n^{2}}\right) = \infty$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(\frac{7}{n^{3}} + \frac{6}{n^{2}}\right) = 13$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(\frac{7}{n^{3}} + \frac{6}{n^{2}}\right) = 13$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(\frac{7}{n^{3}} + \frac{6}{n^{2}}\right) = 0$$
More at n→-oo
The graph