Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (1+3*x)^(5/x)
-
Limit of ((5+x^2-6*x)/(5+x^2-5*x))^(2+3*x)
-
Limit of (2+x^2+3*x)/(-4+x^2)
-
Limit of (-8+x^2+2*x)/(-2+x)
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Identical expressions
- sin(twelve *x)/(three *x)
- sinus of (12 multiply by x) divide by (3 multiply by x)
- sinus of (twelve multiply by x) divide by (three multiply by x)
- sin(12x)/(3x)
- sin12x/3x
- sin(12*x) divide by (3*x)
Limit of the function sin(12*x)/(3*x)
The solution
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[src]
/sin(12*x)\ lim |---------| x->0+\ 3*x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(12 x \right)}}{3 x}\right)$$
Limit(sin(12*x)/((3*x)), x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(12 x \right)}}{3 x}\right)$$
Do replacement
$$u = 12 x$$
then
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(12 x \right)}}{3 x}\right) = \lim_{u \to 0^+}\left(\frac{4 \sin{\left(u \right)}}{u}\right)$$
=
$$4 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
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:
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(12 x \right)}}{3 x}\right) = 4$$
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(12 x \right)}}{3 x}\right)$$
Do replacement
$$u = 12 x$$
then
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(12 x \right)}}{3 x}\right) = \lim_{u \to 0^+}\left(\frac{4 \sin{\left(u \right)}}{u}\right)$$
=
$$4 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
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:
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(12 x \right)}}{3 x}\right) = 4$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(12 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(3 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(12 x \right)}}{3 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(12 x \right)}}{3 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(12 x \right)}}{\frac{d}{d x} 3 x}\right)$$
=
$$\lim_{x \to 0^+}\left(4 \cos{\left(12 x \right)}\right)$$
=
$$\lim_{x \to 0^+} 4$$
=
$$\lim_{x \to 0^+} 4$$
=
$$4$$
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
$$\lim_{x \to 0^+} \sin{\left(12 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(3 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(12 x \right)}}{3 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(12 x \right)}}{3 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(12 x \right)}}{\frac{d}{d x} 3 x}\right)$$
=
$$\lim_{x \to 0^+}\left(4 \cos{\left(12 x \right)}\right)$$
=
$$\lim_{x \to 0^+} 4$$
=
$$\lim_{x \to 0^+} 4$$
=
$$4$$
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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/sin(12*x)\ lim |---------| x->0+\ 3*x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(12 x \right)}}{3 x}\right)$$
4
$$4$$
= 4.0
/sin(12*x)\ lim |---------| x->0-\ 3*x /
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(12 x \right)}}{3 x}\right)$$
4
$$4$$
= 4.0
= 4.0
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(12 x \right)}}{3 x}\right) = 4$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(12 x \right)}}{3 x}\right) = 4$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(12 x \right)}}{3 x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(12 x \right)}}{3 x}\right) = \frac{\sin{\left(12 \right)}}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(12 x \right)}}{3 x}\right) = \frac{\sin{\left(12 \right)}}{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(12 x \right)}}{3 x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(12 x \right)}}{3 x}\right) = 4$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(12 x \right)}}{3 x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(12 x \right)}}{3 x}\right) = \frac{\sin{\left(12 \right)}}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(12 x \right)}}{3 x}\right) = \frac{\sin{\left(12 \right)}}{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(12 x \right)}}{3 x}\right) = 0$$
More at x→-oo
The graph