Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (-2+x^2-x)/(-2+x+3*x^2)
-
Limit of (-1+sin(x))/cos(x)
-
Limit of sin(5*x)/sin(4*x)
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Limit of (-sin(x)+sin(3*x))/x
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Identical expressions
- (-sin(x)+sin(three *x))/x
- ( minus sinus of (x) plus sinus of (3 multiply by x)) divide by x
- ( minus sinus of (x) plus sinus of (three multiply by x)) divide by x
- (-sin(x)+sin(3x))/x
- -sinx+sin3x/x
- (-sin(x)+sin(3*x)) divide by x
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Similar expressions
- (sin(x)+sin(3*x))/x
- (-sin(x)-sin(3*x))/x
- (-sinx+sin(3*x))/x
Limit of the function (-sin(x)+sin(3*x))/x
The solution
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[src]
/-sin(x) + sin(3*x)\ lim |------------------| x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} + \sin{\left(3 x \right)}}{x}\right)$$
Limit((-sin(x) + sin(3*x))/x, x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(- \sin{\left(x \right)} + \sin{\left(3 x \right)}\right) = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} + \sin{\left(3 x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \sin{\left(x \right)} + \sin{\left(3 x \right)}\right)}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(- \cos{\left(x \right)} + 3 \cos{\left(3 x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \cos{\left(x \right)} + 3 \cos{\left(3 x \right)}\right)$$
=
$$2$$
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
$$\lim_{x \to 0^+}\left(- \sin{\left(x \right)} + \sin{\left(3 x \right)}\right) = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} + \sin{\left(3 x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \sin{\left(x \right)} + \sin{\left(3 x \right)}\right)}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(- \cos{\left(x \right)} + 3 \cos{\left(3 x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \cos{\left(x \right)} + 3 \cos{\left(3 x \right)}\right)$$
=
$$2$$
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
$$\lim_{x \to 0^-}\left(\frac{- \sin{\left(x \right)} + \sin{\left(3 x \right)}}{x}\right) = 2$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} + \sin{\left(3 x \right)}}{x}\right) = 2$$
$$\lim_{x \to \infty}\left(\frac{- \sin{\left(x \right)} + \sin{\left(3 x \right)}}{x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{- \sin{\left(x \right)} + \sin{\left(3 x \right)}}{x}\right) = - \sin{\left(1 \right)} + \sin{\left(3 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{- \sin{\left(x \right)} + \sin{\left(3 x \right)}}{x}\right) = - \sin{\left(1 \right)} + \sin{\left(3 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{- \sin{\left(x \right)} + \sin{\left(3 x \right)}}{x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} + \sin{\left(3 x \right)}}{x}\right) = 2$$
$$\lim_{x \to \infty}\left(\frac{- \sin{\left(x \right)} + \sin{\left(3 x \right)}}{x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{- \sin{\left(x \right)} + \sin{\left(3 x \right)}}{x}\right) = - \sin{\left(1 \right)} + \sin{\left(3 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{- \sin{\left(x \right)} + \sin{\left(3 x \right)}}{x}\right) = - \sin{\left(1 \right)} + \sin{\left(3 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{- \sin{\left(x \right)} + \sin{\left(3 x \right)}}{x}\right) = 0$$
More at x→-oo
One‐sided limits
[src]
/-sin(x) + sin(3*x)\ lim |------------------| x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} + \sin{\left(3 x \right)}}{x}\right)$$
2
$$2$$
= 2.0
/-sin(x) + sin(3*x)\ lim |------------------| x->0-\ x /
$$\lim_{x \to 0^-}\left(\frac{- \sin{\left(x \right)} + \sin{\left(3 x \right)}}{x}\right)$$
2
$$2$$
= 2.0
= 2.0
The graph