Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (-2-7*x+4*x^2+5*x^3)/(-3-7*x^2+4*x^3+6*x)
-
Limit of (2-4*x^2+5*x^3)/(3+x)
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Limit of (5^(2*x)-2^(3*x))/(sin(x)+sin(x^2))
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Limit of (1-x^2+4*x)/(-2*x+5*x^2)
- Integral of d{x}:
- cos(x)*sin(x)/x
-
Identical expressions
- cos(x)*sin(x)/x
- co sinus of e of (x) multiply by sinus of (x) divide by x
- cos(x)sin(x)/x
- cosxsinx/x
- cos(x)*sin(x) divide by x
-
Similar expressions
- (1-cos(x))*sin(x)/x
- -(1-cos(x))*sin(x)/x+tan(x)
- (1+1/cos(x))*sin(x)/(x*(1-cos(2*x)))
- cosx*sinx/x
Limit of the function cos(x)*sin(x)/x
The solution
You have entered
[src]
/cos(x)*sin(x)\ lim |-------------| x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x}\right)$$
Limit((cos(x)*sin(x))/x, x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(\sin{\left(x \right)} \cos{\left(x \right)}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(x \right)} \cos{\left(x \right)}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)$$
=
$$1$$
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)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(\sin{\left(x \right)} \cos{\left(x \right)}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(x \right)} \cos{\left(x \right)}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)$$
=
$$1$$
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
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x}\right) = \sin{\left(1 \right)} \cos{\left(1 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x}\right) = \sin{\left(1 \right)} \cos{\left(1 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x}\right) = \sin{\left(1 \right)} \cos{\left(1 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x}\right) = \sin{\left(1 \right)} \cos{\left(1 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x}\right) = 0$$
More at x→-oo
One‐sided limits
[src]
/cos(x)*sin(x)\ lim |-------------| x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x}\right)$$
1
$$1$$
= 1.0
/cos(x)*sin(x)\ lim |-------------| x->0-\ x /
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x}\right)$$
1
$$1$$
= 1.0
= 1.0
The graph