Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (1+x^2+9*x)/(-5+2*x+7*x^2)
-
Limit of (-tan(2*x)+sin(2*x))/x^3
-
Limit of 3/n^4
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Limit of (1-cos(x)^2)/(x^2*sin(x)^2)
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Identical expressions
- (five + two *n)/(one + two *n)
- (5 plus 2 multiply by n) divide by (1 plus 2 multiply by n)
- (five plus two multiply by n) divide by (one plus two multiply by n)
- (5+2n)/(1+2n)
- 5+2n/1+2n
- (5+2*n) divide by (1+2*n)
-
Similar expressions
- -1+((-5+2*n)/(1+2*n))^n
- (5-2*n)/(1+2*n)
- (5+2*n)/(1-2*n)
- ((-5+2*n)/(1+2*n))^(-1+n)
Limit of the function (5+2*n)/(1+2*n)
The solution
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/5 + 2*n\ lim |-------| n->oo\1 + 2*n/
$$\lim_{n \to \infty}\left(\frac{2 n + 5}{2 n + 1}\right)$$
Limit((5 + 2*n)/(1 + 2*n), n, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{n \to \infty}\left(\frac{2 n + 5}{2 n + 1}\right)$$
Let's divide numerator and denominator by n:
$$\lim_{n \to \infty}\left(\frac{2 n + 5}{2 n + 1}\right)$$ =
$$\lim_{n \to \infty}\left(\frac{2 + \frac{5}{n}}{2 + \frac{1}{n}}\right)$$
Do Replacement
$$u = \frac{1}{n}$$
then
$$\lim_{n \to \infty}\left(\frac{2 + \frac{5}{n}}{2 + \frac{1}{n}}\right) = \lim_{u \to 0^+}\left(\frac{5 u + 2}{u + 2}\right)$$
=
$$\frac{0 \cdot 5 + 2}{2} = 1$$
The final answer:
$$\lim_{n \to \infty}\left(\frac{2 n + 5}{2 n + 1}\right) = 1$$
$$\lim_{n \to \infty}\left(\frac{2 n + 5}{2 n + 1}\right)$$
Let's divide numerator and denominator by n:
$$\lim_{n \to \infty}\left(\frac{2 n + 5}{2 n + 1}\right)$$ =
$$\lim_{n \to \infty}\left(\frac{2 + \frac{5}{n}}{2 + \frac{1}{n}}\right)$$
Do Replacement
$$u = \frac{1}{n}$$
then
$$\lim_{n \to \infty}\left(\frac{2 + \frac{5}{n}}{2 + \frac{1}{n}}\right) = \lim_{u \to 0^+}\left(\frac{5 u + 2}{u + 2}\right)$$
=
$$\frac{0 \cdot 5 + 2}{2} = 1$$
The final answer:
$$\lim_{n \to \infty}\left(\frac{2 n + 5}{2 n + 1}\right) = 1$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{n \to \infty}\left(2 n + 5\right) = \infty$$
and limit for the denominator is
$$\lim_{n \to \infty}\left(2 n + 1\right) = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{n \to \infty}\left(\frac{2 n + 5}{2 n + 1}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(2 n + 5\right)}{\frac{d}{d n} \left(2 n + 1\right)}\right)$$
=
$$\lim_{n \to \infty} 1$$
=
$$\lim_{n \to \infty} 1$$
=
$$1$$
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
$$\lim_{n \to \infty}\left(2 n + 5\right) = \infty$$
and limit for the denominator is
$$\lim_{n \to \infty}\left(2 n + 1\right) = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{n \to \infty}\left(\frac{2 n + 5}{2 n + 1}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(2 n + 5\right)}{\frac{d}{d n} \left(2 n + 1\right)}\right)$$
=
$$\lim_{n \to \infty} 1$$
=
$$\lim_{n \to \infty} 1$$
=
$$1$$
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
$$\lim_{n \to \infty}\left(\frac{2 n + 5}{2 n + 1}\right) = 1$$
$$\lim_{n \to 0^-}\left(\frac{2 n + 5}{2 n + 1}\right) = 5$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(\frac{2 n + 5}{2 n + 1}\right) = 5$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(\frac{2 n + 5}{2 n + 1}\right) = \frac{7}{3}$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(\frac{2 n + 5}{2 n + 1}\right) = \frac{7}{3}$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(\frac{2 n + 5}{2 n + 1}\right) = 1$$
More at n→-oo
$$\lim_{n \to 0^-}\left(\frac{2 n + 5}{2 n + 1}\right) = 5$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(\frac{2 n + 5}{2 n + 1}\right) = 5$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(\frac{2 n + 5}{2 n + 1}\right) = \frac{7}{3}$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(\frac{2 n + 5}{2 n + 1}\right) = \frac{7}{3}$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(\frac{2 n + 5}{2 n + 1}\right) = 1$$
More at n→-oo
The graph