Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of sin(2*x)/(5*x)
-
Limit of (1-log(7*x))^(7*x)
-
Limit of (pi/2-atan(x))^(1/x)
-
Limit of sin(3*x)/(7*x)
-
Identical expressions
- sin(three *x)/(seven *x)
- sinus of (3 multiply by x) divide by (7 multiply by x)
- sinus of (three multiply by x) divide by (seven multiply by x)
- sin(3x)/(7x)
- sin3x/7x
- sin(3*x) divide by (7*x)
Limit of the function sin(3*x)/(7*x)
The solution
You have entered
[src]
/sin(3*x)\ lim |--------| x->0+\ 7*x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right)$$
Limit(sin(3*x)/((7*x)), x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right)$$
Do replacement
$$u = 3 x$$
then
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right) = \lim_{u \to 0^+}\left(\frac{3 \sin{\left(u \right)}}{7 u}\right)$$
=
$$\frac{3 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)}{7}$$
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:
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right) = \frac{3}{7}$$
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right)$$
Do replacement
$$u = 3 x$$
then
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right) = \lim_{u \to 0^+}\left(\frac{3 \sin{\left(u \right)}}{7 u}\right)$$
=
$$\frac{3 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)}{7}$$
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:
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right) = \frac{3}{7}$$
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
$$\lim_{x \to 0^+}\left(7 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(3 x \right)}}{\frac{d}{d x} 7 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{3 \cos{\left(3 x \right)}}{7}\right)$$
=
$$\lim_{x \to 0^+} \frac{3}{7}$$
=
$$\lim_{x \to 0^+} \frac{3}{7}$$
=
$$\frac{3}{7}$$
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
$$\lim_{x \to 0^+}\left(7 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(3 x \right)}}{\frac{d}{d x} 7 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{3 \cos{\left(3 x \right)}}{7}\right)$$
=
$$\lim_{x \to 0^+} \frac{3}{7}$$
=
$$\lim_{x \to 0^+} \frac{3}{7}$$
=
$$\frac{3}{7}$$
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
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right) = \frac{3}{7}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right) = \frac{3}{7}$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right) = \frac{\sin{\left(3 \right)}}{7}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right) = \frac{\sin{\left(3 \right)}}{7}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right) = \frac{3}{7}$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right) = \frac{\sin{\left(3 \right)}}{7}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right) = \frac{\sin{\left(3 \right)}}{7}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right) = 0$$
More at x→-oo
One‐sided limits
[src]
/sin(3*x)\ lim |--------| x->0+\ 7*x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right)$$
3/7
$$\frac{3}{7}$$
= 0.428571428571429
/sin(3*x)\ lim |--------| x->0-\ 7*x /
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right)$$
3/7
$$\frac{3}{7}$$
= 0.428571428571429
= 0.428571428571429
The graph