Mister Exam
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How to use it?
- Limit of the function:
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Limit of ((3+x)^2+(3-x)^2)/((3-x)^2-(3+x)^2)
-
Limit of (-1+x)/(-2+sqrt(3+x))
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Limit of (-x+tan(x))/x^3
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Limit of (-1+sqrt(x))/(-3+x)
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Identical expressions
- two *sin(four *x)/x
- 2 multiply by sinus of (4 multiply by x) divide by x
- two multiply by sinus of (four multiply by x) divide by x
- 2sin(4x)/x
- 2sin4x/x
- 2*sin(4*x) divide by x
-
Similar expressions
- (1-cos(x)+2*sin(4*x))/x
Limit of the function 2*sin(4*x)/x
The solution
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/2*sin(4*x)\ lim |----------| x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right)$$
Limit(2*sin(4*x)/x, x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right)$$
Do replacement
$$u = 4 x$$
then
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) = \lim_{u \to 0^+}\left(\frac{8 \sin{\left(u \right)}}{u}\right)$$
=
$$8 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
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:
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) = 8$$
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right)$$
Do replacement
$$u = 4 x$$
then
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) = \lim_{u \to 0^+}\left(\frac{8 \sin{\left(u \right)}}{u}\right)$$
=
$$8 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
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:
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) = 8$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(2 \sin{\left(4 x \right)}\right) = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 2 \sin{\left(4 x \right)}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(8 \cos{\left(4 x \right)}\right)$$
=
$$\lim_{x \to 0^+} 8$$
=
$$\lim_{x \to 0^+} 8$$
=
$$8$$
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
$$\lim_{x \to 0^+}\left(2 \sin{\left(4 x \right)}\right) = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 2 \sin{\left(4 x \right)}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(8 \cos{\left(4 x \right)}\right)$$
=
$$\lim_{x \to 0^+} 8$$
=
$$\lim_{x \to 0^+} 8$$
=
$$8$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) = 8$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) = 8$$
$$\lim_{x \to \infty}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) = 2 \sin{\left(4 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) = 2 \sin{\left(4 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) = 8$$
$$\lim_{x \to \infty}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) = 2 \sin{\left(4 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) = 2 \sin{\left(4 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right) = 0$$
More at x→-oo
One‐sided limits
[src]
/2*sin(4*x)\ lim |----------| x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right)$$
8
$$8$$
= 8
/2*sin(4*x)\ lim |----------| x->0-\ x /
$$\lim_{x \to 0^-}\left(\frac{2 \sin{\left(4 x \right)}}{x}\right)$$
8
$$8$$
= 8
= 8
The graph