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 (3-x^2+5*x)/(4*x^7+81*x)
-
Limit of (-2+2*x^3+7*x)/(-4-x+3*x^3)
-
Identical expressions
- (- five + seven *n)/(two +n)
- ( minus 5 plus 7 multiply by n) divide by (2 plus n)
- ( minus five plus seven multiply by n) divide by (two plus n)
- (-5+7n)/(2+n)
- -5+7n/2+n
- (-5+7*n) divide by (2+n)
-
Similar expressions
- (5+7*n)/(2+n)
- (-5+7*n)/(2-n)
- (-5-7*n)/(2+n)
Limit of the function (-5+7*n)/(2+n)
The solution
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/-5 + 7*n\ lim |--------| n->oo\ 2 + n /
$$\lim_{n \to \infty}\left(\frac{7 n - 5}{n + 2}\right)$$
Limit((-5 + 7*n)/(2 + n), n, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{n \to \infty}\left(\frac{7 n - 5}{n + 2}\right)$$
Let's divide numerator and denominator by n:
$$\lim_{n \to \infty}\left(\frac{7 n - 5}{n + 2}\right)$$ =
$$\lim_{n \to \infty}\left(\frac{7 - \frac{5}{n}}{1 + \frac{2}{n}}\right)$$
Do Replacement
$$u = \frac{1}{n}$$
then
$$\lim_{n \to \infty}\left(\frac{7 - \frac{5}{n}}{1 + \frac{2}{n}}\right) = \lim_{u \to 0^+}\left(\frac{7 - 5 u}{2 u + 1}\right)$$
=
$$\frac{7 - 0}{0 \cdot 2 + 1} = 7$$
The final answer:
$$\lim_{n \to \infty}\left(\frac{7 n - 5}{n + 2}\right) = 7$$
$$\lim_{n \to \infty}\left(\frac{7 n - 5}{n + 2}\right)$$
Let's divide numerator and denominator by n:
$$\lim_{n \to \infty}\left(\frac{7 n - 5}{n + 2}\right)$$ =
$$\lim_{n \to \infty}\left(\frac{7 - \frac{5}{n}}{1 + \frac{2}{n}}\right)$$
Do Replacement
$$u = \frac{1}{n}$$
then
$$\lim_{n \to \infty}\left(\frac{7 - \frac{5}{n}}{1 + \frac{2}{n}}\right) = \lim_{u \to 0^+}\left(\frac{7 - 5 u}{2 u + 1}\right)$$
=
$$\frac{7 - 0}{0 \cdot 2 + 1} = 7$$
The final answer:
$$\lim_{n \to \infty}\left(\frac{7 n - 5}{n + 2}\right) = 7$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{n \to \infty}\left(7 n - 5\right) = \infty$$
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.
$$\lim_{n \to \infty}\left(\frac{7 n - 5}{n + 2}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(7 n - 5\right)}{\frac{d}{d n} \left(n + 2\right)}\right)$$
=
$$\lim_{n \to \infty} 7$$
=
$$\lim_{n \to \infty} 7$$
=
$$7$$
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
$$\lim_{n \to \infty}\left(7 n - 5\right) = \infty$$
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.
$$\lim_{n \to \infty}\left(\frac{7 n - 5}{n + 2}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(7 n - 5\right)}{\frac{d}{d n} \left(n + 2\right)}\right)$$
=
$$\lim_{n \to \infty} 7$$
=
$$\lim_{n \to \infty} 7$$
=
$$7$$
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
$$\lim_{n \to \infty}\left(\frac{7 n - 5}{n + 2}\right) = 7$$
$$\lim_{n \to 0^-}\left(\frac{7 n - 5}{n + 2}\right) = - \frac{5}{2}$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(\frac{7 n - 5}{n + 2}\right) = - \frac{5}{2}$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(\frac{7 n - 5}{n + 2}\right) = \frac{2}{3}$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(\frac{7 n - 5}{n + 2}\right) = \frac{2}{3}$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(\frac{7 n - 5}{n + 2}\right) = 7$$
More at n→-oo
$$\lim_{n \to 0^-}\left(\frac{7 n - 5}{n + 2}\right) = - \frac{5}{2}$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(\frac{7 n - 5}{n + 2}\right) = - \frac{5}{2}$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(\frac{7 n - 5}{n + 2}\right) = \frac{2}{3}$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(\frac{7 n - 5}{n + 2}\right) = \frac{2}{3}$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(\frac{7 n - 5}{n + 2}\right) = 7$$
More at n→-oo
The graph