Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (-16+x^2-6*x)/(-2+x+x^2)
-
Limit of -1+sqrt(5)-x-2/(sqrt(2)-x)
-
Limit of (-cos(x)^3+cos(x))/(4*x*sin(x))
-
Limit of cos((1+x)/x^3)
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Identical expressions
- sin(four *x)/(six *x)
- sinus of (4 multiply by x) divide by (6 multiply by x)
- sinus of (four multiply by x) divide by (six multiply by x)
- sin(4x)/(6x)
- sin4x/6x
- sin(4*x) divide by (6*x)
-
Similar expressions
- (sin(2*x)+sin(4*x))/(6*x)
Limit of the function sin(4*x)/(6*x)
The solution
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[src]
/sin(4*x)\ lim |--------| x->0+\ 6*x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(4 x \right)}}{6 x}\right)$$
Limit(sin(4*x)/((6*x)), x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(4 x \right)}}{6 x}\right)$$
Do replacement
$$u = 4 x$$
then
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(4 x \right)}}{6 x}\right) = \lim_{u \to 0^+}\left(\frac{2 \sin{\left(u \right)}}{3 u}\right)$$
=
$$\frac{2 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)}{3}$$
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:
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(4 x \right)}}{6 x}\right) = \frac{2}{3}$$
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(4 x \right)}}{6 x}\right)$$
Do replacement
$$u = 4 x$$
then
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(4 x \right)}}{6 x}\right) = \lim_{u \to 0^+}\left(\frac{2 \sin{\left(u \right)}}{3 u}\right)$$
=
$$\frac{2 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)}{3}$$
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:
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(4 x \right)}}{6 x}\right) = \frac{2}{3}$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(4 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(6 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(4 x \right)}}{6 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(4 x \right)}}{6 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(4 x \right)}}{\frac{d}{d x} 6 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \cos{\left(4 x \right)}}{3}\right)$$
=
$$\lim_{x \to 0^+} \frac{2}{3}$$
=
$$\lim_{x \to 0^+} \frac{2}{3}$$
=
$$\frac{2}{3}$$
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
$$\lim_{x \to 0^+} \sin{\left(4 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(6 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(4 x \right)}}{6 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(4 x \right)}}{6 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(4 x \right)}}{\frac{d}{d x} 6 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{2 \cos{\left(4 x \right)}}{3}\right)$$
=
$$\lim_{x \to 0^+} \frac{2}{3}$$
=
$$\lim_{x \to 0^+} \frac{2}{3}$$
=
$$\frac{2}{3}$$
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(4*x)\ lim |--------| x->0+\ 6*x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(4 x \right)}}{6 x}\right)$$
2/3
$$\frac{2}{3}$$
= 0.666666666666667
/sin(4*x)\ lim |--------| x->0-\ 6*x /
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(4 x \right)}}{6 x}\right)$$
2/3
$$\frac{2}{3}$$
= 0.666666666666667
= 0.666666666666667
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(4 x \right)}}{6 x}\right) = \frac{2}{3}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(4 x \right)}}{6 x}\right) = \frac{2}{3}$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(4 x \right)}}{6 x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(4 x \right)}}{6 x}\right) = \frac{\sin{\left(4 \right)}}{6}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(4 x \right)}}{6 x}\right) = \frac{\sin{\left(4 \right)}}{6}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(4 x \right)}}{6 x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(4 x \right)}}{6 x}\right) = \frac{2}{3}$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(4 x \right)}}{6 x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(4 x \right)}}{6 x}\right) = \frac{\sin{\left(4 \right)}}{6}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(4 x \right)}}{6 x}\right) = \frac{\sin{\left(4 \right)}}{6}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(4 x \right)}}{6 x}\right) = 0$$
More at x→-oo
The graph