Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
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Limit of sin(log(x))
-
Limit of sin(x)/x
-
Limit of sin(1/z)/(1-z)
- Limit of x/sqrt(x^2+y^2)
- Derivative of:
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sin(x)/x
- Canonical form:
- sin(x)/x
- Graphing y =:
- sin(x)/x
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Identical expressions
- sin(x)/x
- sinus of (x) divide by x
- sinx/x
- sin(x) divide by x
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Similar expressions
- sqrt(x*sin(x))/x^2
- ((x-sin(x))/sin(x))^(sin(x)/(x-sin(x)))
- (-tan(x)+sin(x))/x^3
- (x-sin(x))/(x-tan(x))
- sinx/x
Limit of the function sin(x)/x
The solution
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/sin(x)\ lim |------| x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right)$$
Limit(sin(x)/x, x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right)$$
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right) = \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
=
$$\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
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:
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right) = 1$$
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right)$$
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right) = \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
=
$$\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
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:
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right) = 1$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(x \right)}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+} \cos{\left(x \right)}$$
=
$$\lim_{x \to 0^+} \cos{\left(x \right)}$$
=
$$1$$
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
$$\lim_{x \to 0^+} \sin{\left(x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(x \right)}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+} \cos{\left(x \right)}$$
=
$$\lim_{x \to 0^+} \cos{\left(x \right)}$$
=
$$1$$
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
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)}}{x}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(x \right)}}{x}\right) = \sin{\left(1 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(x \right)}}{x}\right) = \sin{\left(1 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(x \right)}}{x}\right) = \sin{\left(1 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(x \right)}}{x}\right) = \sin{\left(1 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x}\right) = 0$$
More at x→-oo
One‐sided limits
[src]
/sin(x)\ lim |------| x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right)$$
1
$$1$$
= 1.0
/sin(x)\ lim |------| x->0-\ x /
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)}}{x}\right)$$
1
$$1$$
= 1.0
= 1.0
The graph