Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (2+x^2+3*x)/(2+2*x^2+5*x)
-
Limit of (-51+x^2-14*x)/(-21+x^2-4*x)
-
Limit of (28+x^2+11*x)/(-16+x^2)
-
Limit of ((1+x)/(-1+x))^(2+x)
- Derivative of:
- 2^x/(1+x^2)
-
Identical expressions
- two ^x/(one +x^ two)
- 2 to the power of x divide by (1 plus x squared )
- two to the power of x divide by (one plus x to the power of two)
- 2x/(1+x2)
- 2x/1+x2
- 2^x/(1+x²)
- 2 to the power of x/(1+x to the power of 2)
- 2^x/1+x^2
- 2^x divide by (1+x^2)
-
Similar expressions
- 2^x/(1-x^2)
Limit of the function 2^x/(1+x^2)
The solution
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[src]
/ x \
| 2 |
lim |------|
x->oo| 2|
\1 + x /
$$\lim_{x \to \infty}\left(\frac{2^{x}}{x^{2} + 1}\right)$$
Limit(2^x/(1 + x^2), x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \infty} 2^{x} = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty}\left(x^{2} + 1\right) = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{2^{x}}{x^{2} + 1}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} 2^{x}}{\frac{d}{d x} \left(x^{2} + 1\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{2^{x} \log{\left(2 \right)}}{2 x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{2^{x} \log{\left(2 \right)}}{2}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{2^{x} \log{\left(2 \right)}^{2}}{2}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \log{\left(2 \right)}^{2}}{\frac{d}{d x} 2 \cdot 2^{- x}}\right)$$
=
$$\lim_{x \to \infty} 0$$
=
$$\lim_{x \to \infty} 0$$
=
$$\infty$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 3 time(s)
oo/oo,
i.e. limit for the numerator is
$$\lim_{x \to \infty} 2^{x} = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty}\left(x^{2} + 1\right) = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{2^{x}}{x^{2} + 1}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} 2^{x}}{\frac{d}{d x} \left(x^{2} + 1\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{2^{x} \log{\left(2 \right)}}{2 x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{2^{x} \log{\left(2 \right)}}{2}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{2^{x} \log{\left(2 \right)}^{2}}{2}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \log{\left(2 \right)}^{2}}{\frac{d}{d x} 2 \cdot 2^{- x}}\right)$$
=
$$\lim_{x \to \infty} 0$$
=
$$\lim_{x \to \infty} 0$$
=
$$\infty$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 3 time(s)
The graph
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(\frac{2^{x}}{x^{2} + 1}\right) = \infty$$
$$\lim_{x \to 0^-}\left(\frac{2^{x}}{x^{2} + 1}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{2^{x}}{x^{2} + 1}\right) = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{2^{x}}{x^{2} + 1}\right) = 1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{2^{x}}{x^{2} + 1}\right) = 1$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{2^{x}}{x^{2} + 1}\right) = 0$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{2^{x}}{x^{2} + 1}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{2^{x}}{x^{2} + 1}\right) = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{2^{x}}{x^{2} + 1}\right) = 1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{2^{x}}{x^{2} + 1}\right) = 1$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{2^{x}}{x^{2} + 1}\right) = 0$$
More at x→-oo
The graph