Mister Exam
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How to use it?
- Limit of the function:
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Limit of (-log(1+x^2)+sin(x))/(2-e^(3*x)-sqrt(1+x))
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Limit of sin(2*x)/log(1+x)
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Limit of sin(21*x)/(3*x)
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Limit of (-2*sin(x)+sin(2*x))/x
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Identical expressions
- (- two *sin(x)+sin(two *x))/x
- ( minus 2 multiply by sinus of (x) plus sinus of (2 multiply by x)) divide by x
- ( minus two multiply by sinus of (x) plus sinus of (two multiply by x)) divide by x
- (-2sin(x)+sin(2x))/x
- -2sinx+sin2x/x
- (-2*sin(x)+sin(2*x)) divide by x
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Similar expressions
- (-2*sin(x)-sin(2*x))/x
- (-2*sin(x)+sin(2*x))/(x*log(1-x*sin(x)))
- (2*sin(x)+sin(2*x))/x
- (-2*sinx+sin(2*x))/x
Limit of the function (-2*sin(x)+sin(2*x))/x
The solution
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/-2*sin(x) + sin(2*x)\ lim |--------------------| x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{- 2 \sin{\left(x \right)} + \sin{\left(2 x \right)}}{x}\right)$$
Limit((-2*sin(x) + sin(2*x))/x, x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(- 2 \sin{\left(x \right)} + \sin{\left(2 x \right)}\right) = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{- 2 \sin{\left(x \right)} + \sin{\left(2 x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- 2 \sin{\left(x \right)} + \sin{\left(2 x \right)}\right)}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(- 2 \cos{\left(x \right)} + 2 \cos{\left(2 x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- 2 \cos{\left(x \right)} + 2 \cos{\left(2 x \right)}\right)$$
=
$$0$$
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
$$\lim_{x \to 0^+}\left(- 2 \sin{\left(x \right)} + \sin{\left(2 x \right)}\right) = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{- 2 \sin{\left(x \right)} + \sin{\left(2 x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- 2 \sin{\left(x \right)} + \sin{\left(2 x \right)}\right)}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(- 2 \cos{\left(x \right)} + 2 \cos{\left(2 x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- 2 \cos{\left(x \right)} + 2 \cos{\left(2 x \right)}\right)$$
=
$$0$$
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
One‐sided limits
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/-2*sin(x) + sin(2*x)\ lim |--------------------| x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{- 2 \sin{\left(x \right)} + \sin{\left(2 x \right)}}{x}\right)$$
0
$$0$$
= -1.68445193407949e-31
/-2*sin(x) + sin(2*x)\ lim |--------------------| x->0-\ x /
$$\lim_{x \to 0^-}\left(\frac{- 2 \sin{\left(x \right)} + \sin{\left(2 x \right)}}{x}\right)$$
0
$$0$$
= -1.68445193407949e-31
= -1.68445193407949e-31
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{- 2 \sin{\left(x \right)} + \sin{\left(2 x \right)}}{x}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{- 2 \sin{\left(x \right)} + \sin{\left(2 x \right)}}{x}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{- 2 \sin{\left(x \right)} + \sin{\left(2 x \right)}}{x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{- 2 \sin{\left(x \right)} + \sin{\left(2 x \right)}}{x}\right) = - 2 \sin{\left(1 \right)} + \sin{\left(2 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{- 2 \sin{\left(x \right)} + \sin{\left(2 x \right)}}{x}\right) = - 2 \sin{\left(1 \right)} + \sin{\left(2 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{- 2 \sin{\left(x \right)} + \sin{\left(2 x \right)}}{x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{- 2 \sin{\left(x \right)} + \sin{\left(2 x \right)}}{x}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{- 2 \sin{\left(x \right)} + \sin{\left(2 x \right)}}{x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{- 2 \sin{\left(x \right)} + \sin{\left(2 x \right)}}{x}\right) = - 2 \sin{\left(1 \right)} + \sin{\left(2 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{- 2 \sin{\left(x \right)} + \sin{\left(2 x \right)}}{x}\right) = - 2 \sin{\left(1 \right)} + \sin{\left(2 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{- 2 \sin{\left(x \right)} + \sin{\left(2 x \right)}}{x}\right) = 0$$
More at x→-oo
The graph