Mister Exam
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How to use it?
- Limit of the function:
-
Limit of x^(4/x)
-
Limit of (-6+x^2-x)/(-4+x^2)
-
Limit of (-6+x^2-x)/(-21+x+2*x^2)
-
Limit of e^(3-x)*(-2+x)
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Identical expressions
- (x+cos(x))/x
- (x plus co sinus of e of (x)) divide by x
- x+cosx/x
- (x+cos(x)) divide by x
-
Similar expressions
- (-cos(2*x)+cos(x))/x^2
- (-cos(2*x)+cos(x))/(x^3-2*x^2)
- (-sin(x)+cos(x))/(x-pi/4)
- (-cos(2*x)+cos(x))/(x*sin(2*x))
- (-cos(2*x)+cos(x))/(x*sin(3*x))
- (x-cos(x))/x
- (x+cosx)/x
Limit of the function (x+cos(x))/x
The solution
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/x + cos(x)\ lim |----------| x->oo\ x /
$$\lim_{x \to \infty}\left(\frac{x + \cos{\left(x \right)}}{x}\right)$$
Limit((x + cos(x))/x, x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \infty}\left(x + \cos{\left(x \right)}\right) = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty} x = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{x + \cos{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(x + \cos{\left(x \right)}\right)}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to \infty}\left(1 - \sin{\left(x \right)}\right)$$
=
$$\lim_{x \to \infty}\left(1 - \sin{\left(x \right)}\right)$$
=
$$1$$
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
$$\lim_{x \to \infty}\left(x + \cos{\left(x \right)}\right) = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty} x = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{x + \cos{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(x + \cos{\left(x \right)}\right)}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to \infty}\left(1 - \sin{\left(x \right)}\right)$$
=
$$\lim_{x \to \infty}\left(1 - \sin{\left(x \right)}\right)$$
=
$$1$$
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
$$\lim_{x \to \infty}\left(\frac{x + \cos{\left(x \right)}}{x}\right) = 1$$
$$\lim_{x \to 0^-}\left(\frac{x + \cos{\left(x \right)}}{x}\right) = -\infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x + \cos{\left(x \right)}}{x}\right) = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x + \cos{\left(x \right)}}{x}\right) = \cos{\left(1 \right)} + 1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x + \cos{\left(x \right)}}{x}\right) = \cos{\left(1 \right)} + 1$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x + \cos{\left(x \right)}}{x}\right) = 1$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{x + \cos{\left(x \right)}}{x}\right) = -\infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x + \cos{\left(x \right)}}{x}\right) = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x + \cos{\left(x \right)}}{x}\right) = \cos{\left(1 \right)} + 1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x + \cos{\left(x \right)}}{x}\right) = \cos{\left(1 \right)} + 1$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x + \cos{\left(x \right)}}{x}\right) = 1$$
More at x→-oo
The graph