Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (5+x-3*x^2)/(4-x+2*x^2)
-
Limit of (4-x^2)/(3-x^2)
-
Limit of (3+2*x)/(1-5*x)
-
Limit of (1-2*cos(x))/sin(3*x)
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Identical expressions
- sin(five *x)/(two *x^ two)
- sinus of (5 multiply by x) divide by (2 multiply by x squared )
- sinus of (five multiply by x) divide by (two multiply by x to the power of two)
- sin(5*x)/(2*x2)
- sin5*x/2*x2
- sin(5*x)/(2*x²)
- sin(5*x)/(2*x to the power of 2)
- sin(5x)/(2x^2)
- sin(5x)/(2x2)
- sin5x/2x2
- sin5x/2x^2
- sin(5*x) divide by (2*x^2)
Limit of the function sin(5*x)/(2*x^2)
The solution
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[src]
/sin(5*x)\
lim |--------|
x->0+| 2 |
\ 2*x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{2 x^{2}}\right)$$
Limit(sin(5*x)/((2*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
$$\lim_{x \to 0^+}\left(2 x^{2}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{2 x^{2}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{2 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(5 x \right)}}{\frac{d}{d x} 2 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5 \cos{\left(5 x \right)}}{4 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5}{4 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5}{4 x}\right)$$
=
$$\infty$$
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
$$\lim_{x \to 0^+}\left(2 x^{2}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{2 x^{2}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{2 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(5 x \right)}}{\frac{d}{d x} 2 x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5 \cos{\left(5 x \right)}}{4 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5}{4 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5}{4 x}\right)$$
=
$$\infty$$
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
One‐sided limits
[src]
/sin(5*x)\
lim |--------|
x->0+| 2 |
\ 2*x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{2 x^{2}}\right)$$
oo
$$\infty$$
= 377.431019234322
/sin(5*x)\
lim |--------|
x->0-| 2 |
\ 2*x /
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(5 x \right)}}{2 x^{2}}\right)$$
-oo
$$-\infty$$
= -377.431019234322
= -377.431019234322
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(5 x \right)}}{2 x^{2}}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{2 x^{2}}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(5 x \right)}}{2 x^{2}}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(5 x \right)}}{2 x^{2}}\right) = \frac{\sin{\left(5 \right)}}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(5 x \right)}}{2 x^{2}}\right) = \frac{\sin{\left(5 \right)}}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(5 x \right)}}{2 x^{2}}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{2 x^{2}}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(5 x \right)}}{2 x^{2}}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(5 x \right)}}{2 x^{2}}\right) = \frac{\sin{\left(5 \right)}}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(5 x \right)}}{2 x^{2}}\right) = \frac{\sin{\left(5 \right)}}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(5 x \right)}}{2 x^{2}}\right) = 0$$
More at x→-oo
The graph