Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of (-1+e^(3*x))/x
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Limit of sin(8*x)/x
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Limit of cot(3*x)/cot(5*x)
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Limit of (sin(x)+sin(3*x))/(10*x)
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Identical expressions
- (sin(x)+sin(three *x))/(ten *x)
- ( sinus of (x) plus sinus of (3 multiply by x)) divide by (10 multiply by x)
- ( sinus of (x) plus sinus of (three multiply by x)) divide by (ten multiply by x)
- (sin(x)+sin(3x))/(10x)
- sinx+sin3x/10x
- (sin(x)+sin(3*x)) divide by (10*x)
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Similar expressions
- (sin(x)-sin(3*x))/(10*x)
- (sinx+sin(3*x))/(10*x)
Limit of the function (sin(x)+sin(3*x))/(10*x)
The solution
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/sin(x) + sin(3*x)\ lim |-----------------| x->0+\ 10*x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} + \sin{\left(3 x \right)}}{10 x}\right)$$
Limit((sin(x) + sin(3*x))/((10*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
$$\lim_{x \to 0^+}\left(10 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} + \sin{\left(3 x \right)}}{10 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} + \sin{\left(3 x \right)}}{10 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\sin{\left(x \right)} + \sin{\left(3 x \right)}\right)}{\frac{d}{d x} 10 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)}}{10} + \frac{3 \cos{\left(3 x \right)}}{10}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)}}{10} + \frac{3 \cos{\left(3 x \right)}}{10}\right)$$
=
$$\frac{2}{5}$$
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
$$\lim_{x \to 0^+}\left(10 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} + \sin{\left(3 x \right)}}{10 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} + \sin{\left(3 x \right)}}{10 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\sin{\left(x \right)} + \sin{\left(3 x \right)}\right)}{\frac{d}{d x} 10 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)}}{10} + \frac{3 \cos{\left(3 x \right)}}{10}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)}}{10} + \frac{3 \cos{\left(3 x \right)}}{10}\right)$$
=
$$\frac{2}{5}$$
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
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)} + \sin{\left(3 x \right)}}{10 x}\right) = \frac{2}{5}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} + \sin{\left(3 x \right)}}{10 x}\right) = \frac{2}{5}$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} + \sin{\left(3 x \right)}}{10 x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(x \right)} + \sin{\left(3 x \right)}}{10 x}\right) = \frac{\sin{\left(3 \right)}}{10} + \frac{\sin{\left(1 \right)}}{10}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(x \right)} + \sin{\left(3 x \right)}}{10 x}\right) = \frac{\sin{\left(3 \right)}}{10} + \frac{\sin{\left(1 \right)}}{10}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} + \sin{\left(3 x \right)}}{10 x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} + \sin{\left(3 x \right)}}{10 x}\right) = \frac{2}{5}$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} + \sin{\left(3 x \right)}}{10 x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(x \right)} + \sin{\left(3 x \right)}}{10 x}\right) = \frac{\sin{\left(3 \right)}}{10} + \frac{\sin{\left(1 \right)}}{10}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(x \right)} + \sin{\left(3 x \right)}}{10 x}\right) = \frac{\sin{\left(3 \right)}}{10} + \frac{\sin{\left(1 \right)}}{10}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} + \sin{\left(3 x \right)}}{10 x}\right) = 0$$
More at x→-oo
One‐sided limits
[src]
/sin(x) + sin(3*x)\ lim |-----------------| x->0+\ 10*x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} + \sin{\left(3 x \right)}}{10 x}\right)$$
2/5
$$\frac{2}{5}$$
= 0.4
/sin(x) + sin(3*x)\ lim |-----------------| x->0-\ 10*x /
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)} + \sin{\left(3 x \right)}}{10 x}\right)$$
2/5
$$\frac{2}{5}$$
= 0.4
= 0.4
The graph