Mister Exam
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How to use it?
- Limit of the function:
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Limit of (-3*e^(4*x)-2*e^(-x)+5*e^(2*x))/(-4*sqrt(1+5*x)+4*cos(3*x)+5*sin(2*x))
-
Limit of (2+x^2+3*x)/(2+2*x^2+5*x)
- Limit of (-1+x)*cot(pi*(-1+x))
-
Limit of (9+x^10-10*x)/(-1+x)^2
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Identical expressions
- two *sin(three *x)/(three *x)
- 2 multiply by sinus of (3 multiply by x) divide by (3 multiply by x)
- two multiply by sinus of (three multiply by x) divide by (three multiply by x)
- 2sin(3x)/(3x)
- 2sin3x/3x
- 2*sin(3*x) divide by (3*x)
Limit of the function 2*sin(3*x)/(3*x)
The solution
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[src]
/2*sin(3*x)\ lim |----------| x->0+\ 3*x /
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(3 x \right)}}{3 x}\right)$$
Limit((2*sin(3*x))/((3*x)), x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(3 x \right)}}{3 x}\right)$$
Do replacement
$$u = 3 x$$
then
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(3 x \right)}}{3 x}\right) = \lim_{u \to 0^+}\left(\frac{2 \sin{\left(u \right)}}{u}\right)$$
=
$$2 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
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:
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(3 x \right)}}{3 x}\right) = 2$$
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(3 x \right)}}{3 x}\right)$$
Do replacement
$$u = 3 x$$
then
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(3 x \right)}}{3 x}\right) = \lim_{u \to 0^+}\left(\frac{2 \sin{\left(u \right)}}{u}\right)$$
=
$$2 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
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:
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(3 x \right)}}{3 x}\right) = 2$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(3 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(\frac{3 x}{2}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(3 x \right)}}{3 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(3 x \right)}}{3 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(3 x \right)}}{\frac{d}{d x} \frac{3 x}{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(2 \cos{\left(3 x \right)}\right)$$
=
$$\lim_{x \to 0^+} 2$$
=
$$\lim_{x \to 0^+} 2$$
=
$$2$$
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
$$\lim_{x \to 0^+} \sin{\left(3 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(\frac{3 x}{2}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(3 x \right)}}{3 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(3 x \right)}}{3 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(3 x \right)}}{\frac{d}{d x} \frac{3 x}{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(2 \cos{\left(3 x \right)}\right)$$
=
$$\lim_{x \to 0^+} 2$$
=
$$\lim_{x \to 0^+} 2$$
=
$$2$$
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
$$\lim_{x \to 0^-}\left(\frac{2 \sin{\left(3 x \right)}}{3 x}\right) = 2$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(3 x \right)}}{3 x}\right) = 2$$
$$\lim_{x \to \infty}\left(\frac{2 \sin{\left(3 x \right)}}{3 x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{2 \sin{\left(3 x \right)}}{3 x}\right) = \frac{2 \sin{\left(3 \right)}}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{2 \sin{\left(3 x \right)}}{3 x}\right) = \frac{2 \sin{\left(3 \right)}}{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{2 \sin{\left(3 x \right)}}{3 x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(3 x \right)}}{3 x}\right) = 2$$
$$\lim_{x \to \infty}\left(\frac{2 \sin{\left(3 x \right)}}{3 x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{2 \sin{\left(3 x \right)}}{3 x}\right) = \frac{2 \sin{\left(3 \right)}}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{2 \sin{\left(3 x \right)}}{3 x}\right) = \frac{2 \sin{\left(3 \right)}}{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{2 \sin{\left(3 x \right)}}{3 x}\right) = 0$$
More at x→-oo
One‐sided limits
[src]
/2*sin(3*x)\ lim |----------| x->0+\ 3*x /
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(3 x \right)}}{3 x}\right)$$
2
$$2$$
= 2.0
/2*sin(3*x)\ lim |----------| x->0-\ 3*x /
$$\lim_{x \to 0^-}\left(\frac{2 \sin{\left(3 x \right)}}{3 x}\right)$$
2
$$2$$
= 2.0
= 2.0
The graph