Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of sin(5*x)/(7*pi*x)
-
Limit of 3/4
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Limit of (-1+e^(3*x))/x
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Limit of sin(8*x)/x
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Identical expressions
- sin(five *x)/(seven *pi*x)
- sinus of (5 multiply by x) divide by (7 multiply by Pi multiply by x)
- sinus of (five multiply by x) divide by (seven multiply by Pi multiply by x)
- sin(5x)/(7pix)
- sin5x/7pix
- sin(5*x) divide by (7*pi*x)
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What you mean?
- sin(5*x)/((7*pi*x))
- sin(5*x)*x/((7*pi))
- sin(5*x)*pi*x/7
- sin(5*x)/((7*pi*x))
- sin(5*x)*x/((7*pi))
- sin(5*x)*pi*x/7
- sin(5*x)*pi*x/7
You entered:
sin(5*x)/(7*pi*x)
What you mean?
Limit of the function sin(5*x)/(7*pi*x)
The solution
You have entered
[src]
/sin(5*x)\ lim |--------| x->0+\ 7*pi*x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{7 \pi x}\right)$$
Limit(sin(5*x)/((7*pi*x)), x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{7 \pi x}\right)$$
Do replacement
$$u = 5 x$$
then
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{7 \pi x}\right) = \lim_{u \to 0^+}\left(\frac{5 \sin{\left(u \right)}}{7 \pi u}\right)$$
=
$$\frac{5 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)}{7 \pi}$$
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:
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{7 \pi x}\right) = \frac{5}{7 \pi}$$
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{7 \pi x}\right)$$
Do replacement
$$u = 5 x$$
then
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{7 \pi x}\right) = \lim_{u \to 0^+}\left(\frac{5 \sin{\left(u \right)}}{7 \pi u}\right)$$
=
$$\frac{5 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)}{7 \pi}$$
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:
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{7 \pi x}\right) = \frac{5}{7 \pi}$$
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
$$\lim_{x \to 0^+}\left(7 \pi x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{7 \pi x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{7 \pi x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(5 x \right)}}{\frac{d}{d x} 7 \pi x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5 \cos{\left(5 x \right)}}{7 \pi}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5}{7 \pi}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5}{7 \pi}\right)$$
=
$$\frac{5}{7 \pi}$$
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
$$\lim_{x \to 0^+}\left(7 \pi x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{7 \pi x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{7 \pi x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(5 x \right)}}{\frac{d}{d x} 7 \pi x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5 \cos{\left(5 x \right)}}{7 \pi}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5}{7 \pi}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5}{7 \pi}\right)$$
=
$$\frac{5}{7 \pi}$$
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
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(5 x \right)}}{7 \pi x}\right) = \frac{5}{7 \pi}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{7 \pi x}\right) = \frac{5}{7 \pi}$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(5 x \right)}}{7 \pi x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(5 x \right)}}{7 \pi x}\right) = \frac{\sin{\left(5 \right)}}{7 \pi}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(5 x \right)}}{7 \pi x}\right) = \frac{\sin{\left(5 \right)}}{7 \pi}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(5 x \right)}}{7 \pi x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{7 \pi x}\right) = \frac{5}{7 \pi}$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(5 x \right)}}{7 \pi x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(5 x \right)}}{7 \pi x}\right) = \frac{\sin{\left(5 \right)}}{7 \pi}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(5 x \right)}}{7 \pi x}\right) = \frac{\sin{\left(5 \right)}}{7 \pi}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(5 x \right)}}{7 \pi x}\right) = 0$$
More at x→-oo
One‐sided limits
[src]
/sin(5*x)\ lim |--------| x->0+\ 7*pi*x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(5 x \right)}}{7 \pi x}\right)$$
5 ---- 7*pi
$$\frac{5}{7 \pi}$$
= 0.227364204416993
/sin(5*x)\ lim |--------| x->0-\ 7*pi*x /
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(5 x \right)}}{7 \pi x}\right)$$
5 ---- 7*pi
$$\frac{5}{7 \pi}$$
= 0.227364204416993
= 0.227364204416993
The graph