Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (3+x^2-4*x)/(-9+x^2)
-
Limit of (-6+x^2-x)/(9+x^2-6*x)
-
Limit of (-x+tan(x))/(x+2*sin(x))
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Limit of sin(-2+2*x)/(-1+x)
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Identical expressions
- (one + three *n)/(two +n)
- (1 plus 3 multiply by n) divide by (2 plus n)
- (one plus three multiply by n) divide by (two plus n)
- (1+3n)/(2+n)
- 1+3n/2+n
- (1+3*n) divide by (2+n)
-
Similar expressions
- (-1+3*n)/(2+n)
- (1-3*n)/(2+n)
- (1+3*n)/(2-n)
Limit of the function (1+3*n)/(2+n)
The solution
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/1 + 3*n\ lim |-------| n->oo\ 2 + n /
$$\lim_{n \to \infty}\left(\frac{3 n + 1}{n + 2}\right)$$
Limit((1 + 3*n)/(2 + n), n, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{n \to \infty}\left(\frac{3 n + 1}{n + 2}\right)$$
Let's divide numerator and denominator by n:
$$\lim_{n \to \infty}\left(\frac{3 n + 1}{n + 2}\right)$$ =
$$\lim_{n \to \infty}\left(\frac{3 + \frac{1}{n}}{1 + \frac{2}{n}}\right)$$
Do Replacement
$$u = \frac{1}{n}$$
then
$$\lim_{n \to \infty}\left(\frac{3 + \frac{1}{n}}{1 + \frac{2}{n}}\right) = \lim_{u \to 0^+}\left(\frac{u + 3}{2 u + 1}\right)$$
=
$$\frac{3}{0 \cdot 2 + 1} = 3$$
The final answer:
$$\lim_{n \to \infty}\left(\frac{3 n + 1}{n + 2}\right) = 3$$
$$\lim_{n \to \infty}\left(\frac{3 n + 1}{n + 2}\right)$$
Let's divide numerator and denominator by n:
$$\lim_{n \to \infty}\left(\frac{3 n + 1}{n + 2}\right)$$ =
$$\lim_{n \to \infty}\left(\frac{3 + \frac{1}{n}}{1 + \frac{2}{n}}\right)$$
Do Replacement
$$u = \frac{1}{n}$$
then
$$\lim_{n \to \infty}\left(\frac{3 + \frac{1}{n}}{1 + \frac{2}{n}}\right) = \lim_{u \to 0^+}\left(\frac{u + 3}{2 u + 1}\right)$$
=
$$\frac{3}{0 \cdot 2 + 1} = 3$$
The final answer:
$$\lim_{n \to \infty}\left(\frac{3 n + 1}{n + 2}\right) = 3$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{n \to \infty}\left(3 n + 1\right) = \infty$$
and limit for the denominator is
$$\lim_{n \to \infty}\left(n + 2\right) = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{n \to \infty}\left(\frac{3 n + 1}{n + 2}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(3 n + 1\right)}{\frac{d}{d n} \left(n + 2\right)}\right)$$
=
$$\lim_{n \to \infty} 3$$
=
$$\lim_{n \to \infty} 3$$
=
$$3$$
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
$$\lim_{n \to \infty}\left(3 n + 1\right) = \infty$$
and limit for the denominator is
$$\lim_{n \to \infty}\left(n + 2\right) = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{n \to \infty}\left(\frac{3 n + 1}{n + 2}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(3 n + 1\right)}{\frac{d}{d n} \left(n + 2\right)}\right)$$
=
$$\lim_{n \to \infty} 3$$
=
$$\lim_{n \to \infty} 3$$
=
$$3$$
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
$$\lim_{n \to \infty}\left(\frac{3 n + 1}{n + 2}\right) = 3$$
$$\lim_{n \to 0^-}\left(\frac{3 n + 1}{n + 2}\right) = \frac{1}{2}$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(\frac{3 n + 1}{n + 2}\right) = \frac{1}{2}$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(\frac{3 n + 1}{n + 2}\right) = \frac{4}{3}$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(\frac{3 n + 1}{n + 2}\right) = \frac{4}{3}$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(\frac{3 n + 1}{n + 2}\right) = 3$$
More at n→-oo
$$\lim_{n \to 0^-}\left(\frac{3 n + 1}{n + 2}\right) = \frac{1}{2}$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(\frac{3 n + 1}{n + 2}\right) = \frac{1}{2}$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(\frac{3 n + 1}{n + 2}\right) = \frac{4}{3}$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(\frac{3 n + 1}{n + 2}\right) = \frac{4}{3}$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(\frac{3 n + 1}{n + 2}\right) = 3$$
More at n→-oo
The graph