Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of ((-4+3*x)/(2+3*x))^(2*x)
-
Limit of (3+x^2-x)/(-3+3*x+5*x^2)
-
Limit of (-1+x^2)/(-2+x+x^2)
-
Limit of (sqrt(5+x)-sqrt(10))/(-15+x^2-2*x)
- Graphing y =:
- sin(x)^2/tan(x^2)
-
Identical expressions
- sin(x)^ two /tan(x^ two)
- sinus of (x) squared divide by tangent of (x squared )
- sinus of (x) to the power of two divide by tangent of (x to the power of two)
- sin(x)2/tan(x2)
- sinx2/tanx2
- sin(x)²/tan(x²)
- sin(x) to the power of 2/tan(x to the power of 2)
- sinx^2/tanx^2
- sin(x)^2 divide by tan(x^2)
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Similar expressions
- sinx^2/tan(x^2)
Limit of the function sin(x)^2/tan(x^2)
The solution
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[src]
/ 2 \
|sin (x)|
lim |-------|
x->0+| / 2\|
\tan\x //
$$\lim_{x \to 0^+}\left(\frac{\sin^{2}{\left(x \right)}}{\tan{\left(x^{2} \right)}}\right)$$
Limit(sin(x)^2/tan(x^2), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin^{2}{\left(x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \tan{\left(x^{2} \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin^{2}{\left(x \right)}}{\tan{\left(x^{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin^{2}{\left(x \right)}}{\frac{d}{d x} \tan{\left(x^{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x \left(\tan^{2}{\left(x^{2} \right)} + 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 2 \sin{\left(x \right)}}{\frac{d}{d x} 2 x}\right)$$
=
$$\lim_{x \to 0^+} \cos{\left(x \right)}$$
=
$$\lim_{x \to 0^+} 1$$
=
$$\lim_{x \to 0^+} 1$$
=
$$1$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin^{2}{\left(x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \tan{\left(x^{2} \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin^{2}{\left(x \right)}}{\tan{\left(x^{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin^{2}{\left(x \right)}}{\frac{d}{d x} \tan{\left(x^{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} \cos{\left(x \right)}}{x \left(\tan^{2}{\left(x^{2} \right)} + 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 2 \sin{\left(x \right)}}{\frac{d}{d x} 2 x}\right)$$
=
$$\lim_{x \to 0^+} \cos{\left(x \right)}$$
=
$$\lim_{x \to 0^+} 1$$
=
$$\lim_{x \to 0^+} 1$$
=
$$1$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)
The graph
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\sin^{2}{\left(x \right)}}{\tan{\left(x^{2} \right)}}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin^{2}{\left(x \right)}}{\tan{\left(x^{2} \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\sin^{2}{\left(x \right)}}{\tan{\left(x^{2} \right)}}\right) = \left\langle -\infty, \infty\right\rangle$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin^{2}{\left(x \right)}}{\tan{\left(x^{2} \right)}}\right) = \frac{\sin^{2}{\left(1 \right)}}{\tan{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin^{2}{\left(x \right)}}{\tan{\left(x^{2} \right)}}\right) = \frac{\sin^{2}{\left(1 \right)}}{\tan{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(x \right)}}{\tan{\left(x^{2} \right)}}\right) = \left\langle -\infty, \infty\right\rangle$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin^{2}{\left(x \right)}}{\tan{\left(x^{2} \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\sin^{2}{\left(x \right)}}{\tan{\left(x^{2} \right)}}\right) = \left\langle -\infty, \infty\right\rangle$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin^{2}{\left(x \right)}}{\tan{\left(x^{2} \right)}}\right) = \frac{\sin^{2}{\left(1 \right)}}{\tan{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin^{2}{\left(x \right)}}{\tan{\left(x^{2} \right)}}\right) = \frac{\sin^{2}{\left(1 \right)}}{\tan{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(x \right)}}{\tan{\left(x^{2} \right)}}\right) = \left\langle -\infty, \infty\right\rangle$$
More at x→-oo
One‐sided limits
[src]
/ 2 \
|sin (x)|
lim |-------|
x->0+| / 2\|
\tan\x //
$$\lim_{x \to 0^+}\left(\frac{\sin^{2}{\left(x \right)}}{\tan{\left(x^{2} \right)}}\right)$$
1
$$1$$
= 1
/ 2 \
|sin (x)|
lim |-------|
x->0-| / 2\|
\tan\x //
$$\lim_{x \to 0^-}\left(\frac{\sin^{2}{\left(x \right)}}{\tan{\left(x^{2} \right)}}\right)$$
1
$$1$$
= 1
= 1
The graph