Mister Exam
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How to use it?
- Limit of the function:
-
Limit of 2^(-n)*2^(1+n)
- Limit of tan(k*x)/x
-
Limit of (-x+tan(x))/(x+2*sin(x))
-
Limit of asin(5*x)/tan(3*x)
-
Identical expressions
- two ^(-n)* two ^(one +n)
- 2 to the power of ( minus n) multiply by 2 to the power of (1 plus n)
- two to the power of ( minus n) multiply by two to the power of (one plus n)
- 2(-n)*2(1+n)
- 2-n*21+n
- 2^(-n)2^(1+n)
- 2(-n)2(1+n)
- 2-n21+n
- 2^-n2^1+n
-
Similar expressions
- 2^(n)*2^(1+n)
- 2^(-n)*2^(1-n)
Limit of the function 2^(-n)*2^(1+n)
The solution
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/ -n 1 + n\ lim \2 *2 / n->oo
$$\lim_{n \to \infty}\left(2^{- n} 2^{n + 1}\right)$$
Limit(2^(-n)*2^(1 + n), n, oo, dir='-')
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{n \to \infty}\left(2 \cdot 2^{n}\right) = \infty$$
and limit for the denominator is
$$\lim_{n \to \infty} 2^{n} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{n \to \infty}\left(2^{- n} 2^{n + 1}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} 2 \cdot 2^{n}}{\frac{d}{d n} 2^{n}}\right)$$
=
$$\lim_{n \to \infty} 2$$
=
$$\lim_{n \to \infty} 2$$
=
$$2$$
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
$$\lim_{n \to \infty}\left(2 \cdot 2^{n}\right) = \infty$$
and limit for the denominator is
$$\lim_{n \to \infty} 2^{n} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{n \to \infty}\left(2^{- n} 2^{n + 1}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} 2 \cdot 2^{n}}{\frac{d}{d n} 2^{n}}\right)$$
=
$$\lim_{n \to \infty} 2$$
=
$$\lim_{n \to \infty} 2$$
=
$$2$$
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
$$\lim_{n \to \infty}\left(2^{- n} 2^{n + 1}\right) = 2$$
$$\lim_{n \to 0^-}\left(2^{- n} 2^{n + 1}\right) = 2$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(2^{- n} 2^{n + 1}\right) = 2$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(2^{- n} 2^{n + 1}\right) = 2$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(2^{- n} 2^{n + 1}\right) = 2$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(2^{- n} 2^{n + 1}\right) = 2$$
More at n→-oo
$$\lim_{n \to 0^-}\left(2^{- n} 2^{n + 1}\right) = 2$$
More at n→0 from the left
$$\lim_{n \to 0^+}\left(2^{- n} 2^{n + 1}\right) = 2$$
More at n→0 from the right
$$\lim_{n \to 1^-}\left(2^{- n} 2^{n + 1}\right) = 2$$
More at n→1 from the left
$$\lim_{n \to 1^+}\left(2^{- n} 2^{n + 1}\right) = 2$$
More at n→1 from the right
$$\lim_{n \to -\infty}\left(2^{- n} 2^{n + 1}\right) = 2$$
More at n→-oo
The graph