Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of sin(5*x)/(2*x)
-
Limit of (3-x)/(-27+x^3)
-
Limit of (cos(x)+sin(x))^(1/x)
-
Limit of (2/3)^x
-
Identical expressions
- sin(five *x)/(two *x)
- sinus of (5 multiply by x) divide by (2 multiply by x)
- sinus of (five multiply by x) divide by (two multiply by x)
- sin(5x)/(2x)
- sin5x/2x
- sin(5*x) divide by (2*x)
-
Similar expressions
- (sin(x)+sin(5*x))/(2*x)
- sin(5*x)/(2*x^2)
- sin(5*x)/(2*x*cos(4*x))
Limit of the function sin(5*x)/(2*x)
The solution
You have entered
[src]
/sin(5*x)\ lim |--------| x->0+\ 2*x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{2 x}\right)$$
Limit(sin(5*x)/((2*x)), x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{2 x}\right)$$
Do replacement
$$u = 5 x$$
then
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{2 x}\right) = \lim_{u \to 0^+}\left(\frac{5 \sin{\left(u \right)}}{2 u}\right)$$
=
$$\frac{5 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)}{2}$$
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:
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{2 x}\right) = \frac{5}{2}$$
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{2 x}\right)$$
Do replacement
$$u = 5 x$$
then
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{2 x}\right) = \lim_{u \to 0^+}\left(\frac{5 \sin{\left(u \right)}}{2 u}\right)$$
=
$$\frac{5 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)}{2}$$
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:
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{2 x}\right) = \frac{5}{2}$$
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
$$\lim_{x \to 0^+}\left(2 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{2 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{2 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(5 x \right)}}{\frac{d}{d x} 2 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5 \cos{\left(5 x \right)}}{2}\right)$$
=
$$\lim_{x \to 0^+} \frac{5}{2}$$
=
$$\lim_{x \to 0^+} \frac{5}{2}$$
=
$$\frac{5}{2}$$
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
$$\lim_{x \to 0^+} \sin{\left(5 x \right)} = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{2 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{2 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(5 x \right)}}{\frac{d}{d x} 2 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5 \cos{\left(5 x \right)}}{2}\right)$$
=
$$\lim_{x \to 0^+} \frac{5}{2}$$
=
$$\lim_{x \to 0^+} \frac{5}{2}$$
=
$$\frac{5}{2}$$
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
One‐sided limits
[src]
/sin(5*x)\ lim |--------| x->0+\ 2*x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{2 x}\right)$$
5/2
$$\frac{5}{2}$$
= 2.5
/sin(5*x)\ lim |--------| x->0-\ 2*x /
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(5 x \right)}}{2 x}\right)$$
5/2
$$\frac{5}{2}$$
= 2.5
= 2.5
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(5 x \right)}}{2 x}\right) = \frac{5}{2}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{2 x}\right) = \frac{5}{2}$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(5 x \right)}}{2 x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(5 x \right)}}{2 x}\right) = \frac{\sin{\left(5 \right)}}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(5 x \right)}}{2 x}\right) = \frac{\sin{\left(5 \right)}}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(5 x \right)}}{2 x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{2 x}\right) = \frac{5}{2}$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(5 x \right)}}{2 x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(5 x \right)}}{2 x}\right) = \frac{\sin{\left(5 \right)}}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(5 x \right)}}{2 x}\right) = \frac{\sin{\left(5 \right)}}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(5 x \right)}}{2 x}\right) = 0$$
More at x→-oo
The graph