Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of (e^x-e)/(-1+x)
-
Limit of (1-3*x)^(1/x)
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Limit of pi/(x*cot(pi*x/2))
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Limit of sin(7*x)/x^2
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Identical expressions
- sin(seven *x)/x^ two
- sinus of (7 multiply by x) divide by x squared
- sinus of (seven multiply by x) divide by x to the power of two
- sin(7*x)/x2
- sin7*x/x2
- sin(7*x)/x²
- sin(7*x)/x to the power of 2
- sin(7x)/x^2
- sin(7x)/x2
- sin7x/x2
- sin7x/x^2
- sin(7*x) divide by x^2
Limit of the function sin(7*x)/x^2
The solution
You have entered
[src]
/sin(7*x)\
lim |--------|
x->0+| 2 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{x^{2}}\right)$$
Limit(sin(7*x)/x^2, x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(7 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x^{2} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(7 x \right)}}{\frac{d}{d x} x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{7 \cos{\left(7 x \right)}}{2 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{7}{2 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{7}{2 x}\right)$$
=
$$\infty$$
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
$$\lim_{x \to 0^+} \sin{\left(7 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x^{2} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(7 x \right)}}{\frac{d}{d x} x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{7 \cos{\left(7 x \right)}}{2 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{7}{2 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{7}{2 x}\right)$$
=
$$\infty$$
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(7*x)\
lim |--------|
x->0+| 2 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{x^{2}}\right)$$
oo
$$\infty$$
= 1056.62145348119
/sin(7*x)\
lim |--------|
x->0-| 2 |
\ x /
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(7 x \right)}}{x^{2}}\right)$$
-oo
$$-\infty$$
= -1056.62145348119
= -1056.62145348119
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(7 x \right)}}{x^{2}}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{x^{2}}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(7 x \right)}}{x^{2}}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(7 x \right)}}{x^{2}}\right) = \sin{\left(7 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(7 x \right)}}{x^{2}}\right) = \sin{\left(7 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(7 x \right)}}{x^{2}}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{x^{2}}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(7 x \right)}}{x^{2}}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(7 x \right)}}{x^{2}}\right) = \sin{\left(7 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(7 x \right)}}{x^{2}}\right) = \sin{\left(7 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(7 x \right)}}{x^{2}}\right) = 0$$
More at x→-oo
The graph