Mister Exam
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How to use it?
- Limit of the function:
-
Limit of ((-2+x)/(1+x))^(-3+2*x)
-
Limit of (-2*asin(x)+asin(2*x))/x^3
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Limit of -cos(x)+5*x
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Limit of sin(5*x)/(4*x^2)
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Identical expressions
- sin(five *x)/(four *x^ two)
- sinus of (5 multiply by x) divide by (4 multiply by x squared )
- sinus of (five multiply by x) divide by (four multiply by x to the power of two)
- sin(5*x)/(4*x2)
- sin5*x/4*x2
- sin(5*x)/(4*x²)
- sin(5*x)/(4*x to the power of 2)
- sin(5x)/(4x^2)
- sin(5x)/(4x2)
- sin5x/4x2
- sin5x/4x^2
- sin(5*x) divide by (4*x^2)
Limit of the function sin(5*x)/(4*x^2)
The solution
You have entered
[src]
/sin(5*x)\
lim |--------|
x->0+| 2 |
\ 4*x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{4 x^{2}}\right)$$
Limit(sin(5*x)/((4*x^2)), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(5 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(4 x^{2}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{4 x^{2}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{4 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(5 x \right)}}{\frac{d}{d x} 4 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5 \cos{\left(5 x \right)}}{8 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5}{8 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5}{8 x}\right)$$
=
$$\infty$$
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
$$\lim_{x \to 0^+} \sin{\left(5 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(4 x^{2}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{4 x^{2}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{4 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(5 x \right)}}{\frac{d}{d x} 4 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5 \cos{\left(5 x \right)}}{8 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5}{8 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5}{8 x}\right)$$
=
$$\infty$$
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
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(5 x \right)}}{4 x^{2}}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{4 x^{2}}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(5 x \right)}}{4 x^{2}}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(5 x \right)}}{4 x^{2}}\right) = \frac{\sin{\left(5 \right)}}{4}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(5 x \right)}}{4 x^{2}}\right) = \frac{\sin{\left(5 \right)}}{4}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(5 x \right)}}{4 x^{2}}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{4 x^{2}}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(5 x \right)}}{4 x^{2}}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(5 x \right)}}{4 x^{2}}\right) = \frac{\sin{\left(5 \right)}}{4}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(5 x \right)}}{4 x^{2}}\right) = \frac{\sin{\left(5 \right)}}{4}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(5 x \right)}}{4 x^{2}}\right) = 0$$
More at x→-oo
One‐sided limits
[src]
/sin(5*x)\
lim |--------|
x->0+| 2 |
\ 4*x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{4 x^{2}}\right)$$
oo
$$\infty$$
= 188.715509617161
/sin(5*x)\
lim |--------|
x->0-| 2 |
\ 4*x /
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(5 x \right)}}{4 x^{2}}\right)$$
-oo
$$-\infty$$
= -188.715509617161
= -188.715509617161
The graph