Mister Exam
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How to use it?
- Limit of the function:
- Limit of (1/factorial(x))^(1/x)
- Limit of (1+cos(3*x))^(1/cos(x))
-
Limit of (1-cos(2*x))/(1-cos(4*x))
-
Limit of (1-4*x)/(-2+5*x)
-
Identical expressions
- -sin(six *x)/(two *x)
- minus sinus of (6 multiply by x) divide by (2 multiply by x)
- minus sinus of (six multiply by x) divide by (two multiply by x)
- -sin(6x)/(2x)
- -sin6x/2x
- -sin(6*x) divide by (2*x)
-
Similar expressions
- sin(6*x)/(2*x)
Limit of the function -sin(6*x)/(2*x)
The solution
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/-sin(6*x) \ lim |----------| x->0+\ 2*x /
$$\lim_{x \to 0^+}\left(\frac{\left(-1\right) \sin{\left(6 x \right)}}{2 x}\right)$$
Limit((-sin(6*x))/((2*x)), x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{\left(-1\right) \sin{\left(6 x \right)}}{2 x}\right)$$
Do replacement
$$u = 6 x$$
then
$$\lim_{x \to 0^+}\left(- \frac{\sin{\left(6 x \right)}}{2 x}\right) = \lim_{u \to 0^+}\left(- \frac{3 \sin{\left(u \right)}}{u}\right)$$
=
$$- 3 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
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:
$$\lim_{x \to 0^+}\left(\frac{\left(-1\right) \sin{\left(6 x \right)}}{2 x}\right) = -3$$
$$\lim_{x \to 0^+}\left(\frac{\left(-1\right) \sin{\left(6 x \right)}}{2 x}\right)$$
Do replacement
$$u = 6 x$$
then
$$\lim_{x \to 0^+}\left(- \frac{\sin{\left(6 x \right)}}{2 x}\right) = \lim_{u \to 0^+}\left(- \frac{3 \sin{\left(u \right)}}{u}\right)$$
=
$$- 3 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
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:
$$\lim_{x \to 0^+}\left(\frac{\left(-1\right) \sin{\left(6 x \right)}}{2 x}\right) = -3$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(6 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(- 2 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\left(-1\right) \sin{\left(6 x \right)}}{2 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(- \frac{\sin{\left(6 x \right)}}{2 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(6 x \right)}}{\frac{d}{d x} \left(- 2 x\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- 3 \cos{\left(6 x \right)}\right)$$
=
$$\lim_{x \to 0^+} -3$$
=
$$\lim_{x \to 0^+} -3$$
=
$$-3$$
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
$$\lim_{x \to 0^+} \sin{\left(6 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(- 2 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\left(-1\right) \sin{\left(6 x \right)}}{2 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(- \frac{\sin{\left(6 x \right)}}{2 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(6 x \right)}}{\frac{d}{d x} \left(- 2 x\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- 3 \cos{\left(6 x \right)}\right)$$
=
$$\lim_{x \to 0^+} -3$$
=
$$\lim_{x \to 0^+} -3$$
=
$$-3$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\left(-1\right) \sin{\left(6 x \right)}}{2 x}\right) = -3$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\left(-1\right) \sin{\left(6 x \right)}}{2 x}\right) = -3$$
$$\lim_{x \to \infty}\left(\frac{\left(-1\right) \sin{\left(6 x \right)}}{2 x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\left(-1\right) \sin{\left(6 x \right)}}{2 x}\right) = - \frac{\sin{\left(6 \right)}}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\left(-1\right) \sin{\left(6 x \right)}}{2 x}\right) = - \frac{\sin{\left(6 \right)}}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\left(-1\right) \sin{\left(6 x \right)}}{2 x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\left(-1\right) \sin{\left(6 x \right)}}{2 x}\right) = -3$$
$$\lim_{x \to \infty}\left(\frac{\left(-1\right) \sin{\left(6 x \right)}}{2 x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\left(-1\right) \sin{\left(6 x \right)}}{2 x}\right) = - \frac{\sin{\left(6 \right)}}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\left(-1\right) \sin{\left(6 x \right)}}{2 x}\right) = - \frac{\sin{\left(6 \right)}}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\left(-1\right) \sin{\left(6 x \right)}}{2 x}\right) = 0$$
More at x→-oo
One‐sided limits
[src]
/-sin(6*x) \ lim |----------| x->0+\ 2*x /
$$\lim_{x \to 0^+}\left(\frac{\left(-1\right) \sin{\left(6 x \right)}}{2 x}\right)$$
-3
$$-3$$
= -3.0
/-sin(6*x) \ lim |----------| x->0-\ 2*x /
$$\lim_{x \to 0^-}\left(\frac{\left(-1\right) \sin{\left(6 x \right)}}{2 x}\right)$$
-3
$$-3$$
= -3.0
= -3.0
The graph