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(eight *x)/x
- sinus of (8 multiply by x) divide by x
- sinus of (eight multiply by x) divide by x
- sin(8x)/x
- sin8x/x
- sin(8*x) divide by x
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Similar expressions
- sin(8*x)/(x*tan(4*x))
- sin(8*x)/(x^2+pi*x)
- sin(8*x)/(x*cot(3*x)*tan(5)^2)
Limit of the function sin(8*x)/x
The solution
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[src]
/sin(8*x)\ lim |--------| x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(8 x \right)}}{x}\right)$$
Limit(sin(8*x)/x, x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(8 x \right)}}{x}\right)$$
Do replacement
$$u = 8 x$$
then
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(8 x \right)}}{x}\right) = \lim_{u \to 0^+}\left(\frac{8 \sin{\left(u \right)}}{u}\right)$$
=
$$8 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
The limit
$$\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
is first remarkable limit, is equal to 1.
The final answer:
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(8 x \right)}}{x}\right) = 8$$
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(8 x \right)}}{x}\right)$$
Do replacement
$$u = 8 x$$
then
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(8 x \right)}}{x}\right) = \lim_{u \to 0^+}\left(\frac{8 \sin{\left(u \right)}}{u}\right)$$
=
$$8 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
The limit
$$\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
is first remarkable limit, is equal to 1.
The final answer:
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(8 x \right)}}{x}\right) = 8$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(8 x \right)} = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(8 x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(8 x \right)}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(8 \cos{\left(8 x \right)}\right)$$
=
$$\lim_{x \to 0^+} 8$$
=
$$\lim_{x \to 0^+} 8$$
=
$$8$$
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
$$\lim_{x \to 0^+} \sin{\left(8 x \right)} = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(8 x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(8 x \right)}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(8 \cos{\left(8 x \right)}\right)$$
=
$$\lim_{x \to 0^+} 8$$
=
$$\lim_{x \to 0^+} 8$$
=
$$8$$
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
[src]
/sin(8*x)\ lim |--------| x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(8 x \right)}}{x}\right)$$
8
$$8$$
= 8.0
/sin(8*x)\ lim |--------| x->0-\ x /
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(8 x \right)}}{x}\right)$$
8
$$8$$
= 8.0
= 8.0
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(8 x \right)}}{x}\right) = 8$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(8 x \right)}}{x}\right) = 8$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(8 x \right)}}{x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(8 x \right)}}{x}\right) = \sin{\left(8 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(8 x \right)}}{x}\right) = \sin{\left(8 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(8 x \right)}}{x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(8 x \right)}}{x}\right) = 8$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(8 x \right)}}{x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(8 x \right)}}{x}\right) = \sin{\left(8 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(8 x \right)}}{x}\right) = \sin{\left(8 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(8 x \right)}}{x}\right) = 0$$
More at x→-oo
The graph