Mister Exam
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How to use it?
- Limit of the function:
-
Limit of (-2+2*x^2+log(x))/(e^x-e)
-
Limit of (-x^2+4*x)/(2-sqrt(x))
-
Limit of ((2+3*x)/(-1+3*x))^(-1+4*x)
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Limit of (3-x+2*x^2)/(5+x^3-8*x)
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Identical expressions
- sin(x)*tan(x/ two)
- sinus of (x) multiply by tangent of (x divide by 2)
- sinus of (x) multiply by tangent of (x divide by two)
- sin(x)tan(x/2)
- sinxtanx/2
- sin(x)*tan(x divide by 2)
-
Similar expressions
- x*(2+x)^3*log(1+x*sin(x)*tan(x))/((2^x-3^x)*sin(5*x)*tan(3*x^2))
- x-sin(x)*tan(x)/(2*x^3)
- sinx*tan(x/2)
Limit of the function sin(x)*tan(x/2)
The solution
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[src]
/ /x\\ lim |sin(x)*tan|-|| x->pi+\ \2//
$$\lim_{x \to \pi^+}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right)$$
Limit(sin(x)*tan(x/2), x, pi)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \pi^+} \sin{\left(x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to \pi^+} \frac{1}{\tan{\left(\frac{x}{2} \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \pi^+}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right)$$
=
$$\lim_{x \to \pi^+}\left(\frac{\frac{d}{d x} \sin{\left(x \right)}}{\frac{d}{d x} \frac{1}{\tan{\left(\frac{x}{2} \right)}}}\right)$$
=
$$\lim_{x \to \pi^+}\left(\frac{\cos{\left(x \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{- \frac{\tan^{2}{\left(\frac{x}{2} \right)}}{2} - \frac{1}{2}}\right)$$
=
$$\lim_{x \to \pi^+}\left(\frac{\cos{\left(x \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{- \frac{\tan^{2}{\left(\frac{x}{2} \right)}}{2} - \frac{1}{2}}\right)$$
=
$$2$$
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
$$\lim_{x \to \pi^+} \sin{\left(x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to \pi^+} \frac{1}{\tan{\left(\frac{x}{2} \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \pi^+}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right)$$
=
$$\lim_{x \to \pi^+}\left(\frac{\frac{d}{d x} \sin{\left(x \right)}}{\frac{d}{d x} \frac{1}{\tan{\left(\frac{x}{2} \right)}}}\right)$$
=
$$\lim_{x \to \pi^+}\left(\frac{\cos{\left(x \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{- \frac{\tan^{2}{\left(\frac{x}{2} \right)}}{2} - \frac{1}{2}}\right)$$
=
$$\lim_{x \to \pi^+}\left(\frac{\cos{\left(x \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{- \frac{\tan^{2}{\left(\frac{x}{2} \right)}}{2} - \frac{1}{2}}\right)$$
=
$$2$$
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]
/ /x\\ lim |sin(x)*tan|-|| x->pi+\ \2//
$$\lim_{x \to \pi^+}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right)$$
2
$$2$$
= 2.0
/ /x\\ lim |sin(x)*tan|-|| x->pi-\ \2//
$$\lim_{x \to \pi^-}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right)$$
2
$$2$$
= 2.0
= 2.0
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \pi^-}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right) = 2$$
More at x→pi from the left
$$\lim_{x \to \pi^+}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right) = 2$$
$$\lim_{x \to \infty}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right)$$
More at x→oo
$$\lim_{x \to 0^-}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right) = \sin{\left(1 \right)} \tan{\left(\frac{1}{2} \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right) = \sin{\left(1 \right)} \tan{\left(\frac{1}{2} \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right)$$
More at x→-oo
More at x→pi from the left
$$\lim_{x \to \pi^+}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right) = 2$$
$$\lim_{x \to \infty}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right)$$
More at x→oo
$$\lim_{x \to 0^-}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right) = \sin{\left(1 \right)} \tan{\left(\frac{1}{2} \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right) = \sin{\left(1 \right)} \tan{\left(\frac{1}{2} \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\sin{\left(x \right)} \tan{\left(\frac{x}{2} \right)}\right)$$
More at x→-oo
The graph