Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (1+3*x)^(5/x)
-
Limit of (-16+x^2+6*x)/(-2-5*x+3*x^2)
-
Limit of (1+x)^(2/3)-(-1+x)^(2/3)
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Limit of e^(1+3*x)*(-1+x)
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Identical expressions
- (x+ one /x)/x
- (x plus 1 divide by x) divide by x
- (x plus one divide by x) divide by x
- x+1/x/x
- (x+1 divide by x) divide by x
-
Similar expressions
- (x-1/x)/x
Limit of the function (x+1/x)/x
The solution
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/ 1\
|x + -|
| x|
lim |-----|
x->-oo\ x /
$$\lim_{x \to -\infty}\left(\frac{x + \frac{1}{x}}{x}\right)$$
Limit((x + 1/x)/x, x, -oo)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to -\infty}\left(x^{2} + 1\right) = \infty$$
and limit for the denominator is
$$\lim_{x \to -\infty} x^{2} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to -\infty}\left(\frac{x + \frac{1}{x}}{x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to -\infty}\left(\frac{x^{2} + 1}{x^{2}}\right)$$
=
$$\lim_{x \to -\infty}\left(\frac{\frac{d}{d x} \left(x^{2} + 1\right)}{\frac{d}{d x} x^{2}}\right)$$
=
$$\lim_{x \to -\infty} 1$$
=
$$\lim_{x \to -\infty} 1$$
=
$$1$$
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
$$\lim_{x \to -\infty}\left(x^{2} + 1\right) = \infty$$
and limit for the denominator is
$$\lim_{x \to -\infty} x^{2} = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to -\infty}\left(\frac{x + \frac{1}{x}}{x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to -\infty}\left(\frac{x^{2} + 1}{x^{2}}\right)$$
=
$$\lim_{x \to -\infty}\left(\frac{\frac{d}{d x} \left(x^{2} + 1\right)}{\frac{d}{d x} x^{2}}\right)$$
=
$$\lim_{x \to -\infty} 1$$
=
$$\lim_{x \to -\infty} 1$$
=
$$1$$
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 x→0, -oo, +oo, 1
$$\lim_{x \to -\infty}\left(\frac{x + \frac{1}{x}}{x}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{x + \frac{1}{x}}{x}\right) = 1$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{x + \frac{1}{x}}{x}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x + \frac{1}{x}}{x}\right) = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x + \frac{1}{x}}{x}\right) = 2$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x + \frac{1}{x}}{x}\right) = 2$$
More at x→1 from the right
$$\lim_{x \to \infty}\left(\frac{x + \frac{1}{x}}{x}\right) = 1$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{x + \frac{1}{x}}{x}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x + \frac{1}{x}}{x}\right) = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x + \frac{1}{x}}{x}\right) = 2$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x + \frac{1}{x}}{x}\right) = 2$$
More at x→1 from the right
The graph