Mister Exam
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- Limit of the function:
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Limit of -cos(x)+5*x
-
Limit of (-30+x^2-x)/(125+x^3)
-
Limit of sin(9*x)
- Limit of cot(pi*x)
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Identical expressions
- sin(two *x)/(two *tan(x))
- sinus of (2 multiply by x) divide by (2 multiply by tangent of (x))
- sinus of (two multiply by x) divide by (two multiply by tangent of (x))
- sin(2x)/(2tan(x))
- sin2x/2tanx
- sin(2*x) divide by (2*tan(x))
Limit of the function sin(2*x)/(2*tan(x))
The solution
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/sin(2*x)\ lim |--------| x->pi+\2*tan(x)/
$$\lim_{x \to \pi^+}\left(\frac{\sin{\left(2 x \right)}}{2 \tan{\left(x \right)}}\right)$$
Limit(sin(2*x)/((2*tan(x))), x, pi)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \pi^+}\left(\frac{\sin{\left(2 x \right)}}{2}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to \pi^+} \tan{\left(x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \pi^+}\left(\frac{\sin{\left(2 x \right)}}{2 \tan{\left(x \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to \pi^+}\left(\frac{\sin{\left(2 x \right)}}{2 \tan{\left(x \right)}}\right)$$
=
$$\lim_{x \to \pi^+}\left(\frac{\frac{d}{d x} \frac{\sin{\left(2 x \right)}}{2}}{\frac{d}{d x} \tan{\left(x \right)}}\right)$$
=
$$\lim_{x \to \pi^+}\left(\frac{\cos{\left(2 x \right)}}{\tan^{2}{\left(x \right)} + 1}\right)$$
=
$$\lim_{x \to \pi^+}\left(\frac{\cos{\left(2 x \right)}}{\tan^{2}{\left(x \right)} + 1}\right)$$
=
$$1$$
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
$$\lim_{x \to \pi^+}\left(\frac{\sin{\left(2 x \right)}}{2}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to \pi^+} \tan{\left(x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \pi^+}\left(\frac{\sin{\left(2 x \right)}}{2 \tan{\left(x \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to \pi^+}\left(\frac{\sin{\left(2 x \right)}}{2 \tan{\left(x \right)}}\right)$$
=
$$\lim_{x \to \pi^+}\left(\frac{\frac{d}{d x} \frac{\sin{\left(2 x \right)}}{2}}{\frac{d}{d x} \tan{\left(x \right)}}\right)$$
=
$$\lim_{x \to \pi^+}\left(\frac{\cos{\left(2 x \right)}}{\tan^{2}{\left(x \right)} + 1}\right)$$
=
$$\lim_{x \to \pi^+}\left(\frac{\cos{\left(2 x \right)}}{\tan^{2}{\left(x \right)} + 1}\right)$$
=
$$1$$
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
One‐sided limits
[src]
/sin(2*x)\ lim |--------| x->pi+\2*tan(x)/
$$\lim_{x \to \pi^+}\left(\frac{\sin{\left(2 x \right)}}{2 \tan{\left(x \right)}}\right)$$
1
$$1$$
= 1.0
/sin(2*x)\ lim |--------| x->pi-\2*tan(x)/
$$\lim_{x \to \pi^-}\left(\frac{\sin{\left(2 x \right)}}{2 \tan{\left(x \right)}}\right)$$
1
$$1$$
= 1.0
= 1.0
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \pi^-}\left(\frac{\sin{\left(2 x \right)}}{2 \tan{\left(x \right)}}\right) = 1$$
More at x→pi from the left
$$\lim_{x \to \pi^+}\left(\frac{\sin{\left(2 x \right)}}{2 \tan{\left(x \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)}}{2 \tan{\left(x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(2 x \right)}}{2 \tan{\left(x \right)}}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(2 x \right)}}{2 \tan{\left(x \right)}}\right) = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(2 x \right)}}{2 \tan{\left(x \right)}}\right) = \frac{\sin{\left(2 \right)}}{2 \tan{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(2 x \right)}}{2 \tan{\left(x \right)}}\right) = \frac{\sin{\left(2 \right)}}{2 \tan{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)}}{2 \tan{\left(x \right)}}\right)$$
More at x→-oo
More at x→pi from the left
$$\lim_{x \to \pi^+}\left(\frac{\sin{\left(2 x \right)}}{2 \tan{\left(x \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)}}{2 \tan{\left(x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(2 x \right)}}{2 \tan{\left(x \right)}}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(2 x \right)}}{2 \tan{\left(x \right)}}\right) = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(2 x \right)}}{2 \tan{\left(x \right)}}\right) = \frac{\sin{\left(2 \right)}}{2 \tan{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(2 x \right)}}{2 \tan{\left(x \right)}}\right) = \frac{\sin{\left(2 \right)}}{2 \tan{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)}}{2 \tan{\left(x \right)}}\right)$$
More at x→-oo
The graph