Mister Exam
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- Limit of the function:
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Limit of (-3*e^(4*x)-2*e^(-x)+5*e^(2*x))/(-4*sqrt(1+5*x)+4*cos(3*x)+5*sin(2*x))
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Limit of (2+x^2+3*x)/(2+2*x^2+5*x)
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Limit of ((-1+x)/(2+x))^(1+2*x)
- Limit of (-1+x)*cot(pi*(-1+x))
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Identical expressions
- (- one +x)*cot(pi*(- one +x))
- ( minus 1 plus x) multiply by cotangent of ( Pi multiply by ( minus 1 plus x))
- ( minus one plus x) multiply by cotangent of ( Pi multiply by ( minus one plus x))
- (-1+x)cot(pi(-1+x))
- -1+xcotpi-1+x
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Similar expressions
- (1+x)*cot(pi*(-1+x))
- (-1-x)*cot(pi*(-1+x))
- (-1+x)*cot(pi*(-1-x))
- (-1+x)*cot(pi*(1+x))
Limit of the function (-1+x)*cot(pi*(-1+x))
The solution
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lim ((-1 + x)*cot(pi*(-1 + x))) x->1+
$$\lim_{x \to 1^+}\left(\left(x - 1\right) \cot{\left(\pi \left(x - 1\right) \right)}\right)$$
Limit((-1 + x)*cot(pi*(-1 + x)), x, 1)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 1^+}\left(x - 1\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 1^+} \frac{1}{\cot{\left(\pi \left(x - 1\right) \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 1^+}\left(\left(x - 1\right) \cot{\left(\pi \left(x - 1\right) \right)}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(x - 1\right)}{\frac{d}{d x} \frac{1}{\cot{\left(\pi \left(x - 1\right) \right)}}}\right)$$
=
$$\lim_{x \to 1^+}\left(- \frac{\cot^{2}{\left(\pi x \right)}}{\pi \left(- \cot^{2}{\left(\pi x \right)} - 1\right)}\right)$$
=
$$\lim_{x \to 1^+}\left(- \frac{\cot^{2}{\left(\pi x \right)}}{\pi \left(- \cot^{2}{\left(\pi x \right)} - 1\right)}\right)$$
=
$$\frac{1}{\pi}$$
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
$$\lim_{x \to 1^+}\left(x - 1\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 1^+} \frac{1}{\cot{\left(\pi \left(x - 1\right) \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 1^+}\left(\left(x - 1\right) \cot{\left(\pi \left(x - 1\right) \right)}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(x - 1\right)}{\frac{d}{d x} \frac{1}{\cot{\left(\pi \left(x - 1\right) \right)}}}\right)$$
=
$$\lim_{x \to 1^+}\left(- \frac{\cot^{2}{\left(\pi x \right)}}{\pi \left(- \cot^{2}{\left(\pi x \right)} - 1\right)}\right)$$
=
$$\lim_{x \to 1^+}\left(- \frac{\cot^{2}{\left(\pi x \right)}}{\pi \left(- \cot^{2}{\left(\pi x \right)} - 1\right)}\right)$$
=
$$\frac{1}{\pi}$$
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)
One‐sided limits
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lim ((-1 + x)*cot(pi*(-1 + x))) x->1+
$$\lim_{x \to 1^+}\left(\left(x - 1\right) \cot{\left(\pi \left(x - 1\right) \right)}\right)$$
1 -- pi
$$\frac{1}{\pi}$$
= 0.318309886183791
lim ((-1 + x)*cot(pi*(-1 + x))) x->1-
$$\lim_{x \to 1^-}\left(\left(x - 1\right) \cot{\left(\pi \left(x - 1\right) \right)}\right)$$
1 -- pi
$$\frac{1}{\pi}$$
= 0.318309886183791
= 0.318309886183791
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 1^-}\left(\left(x - 1\right) \cot{\left(\pi \left(x - 1\right) \right)}\right) = \frac{1}{\pi}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\left(x - 1\right) \cot{\left(\pi \left(x - 1\right) \right)}\right) = \frac{1}{\pi}$$
$$\lim_{x \to \infty}\left(\left(x - 1\right) \cot{\left(\pi \left(x - 1\right) \right)}\right)$$
More at x→oo
$$\lim_{x \to 0^-}\left(\left(x - 1\right) \cot{\left(\pi \left(x - 1\right) \right)}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\left(x - 1\right) \cot{\left(\pi \left(x - 1\right) \right)}\right) = -\infty$$
More at x→0 from the right
$$\lim_{x \to -\infty}\left(\left(x - 1\right) \cot{\left(\pi \left(x - 1\right) \right)}\right)$$
More at x→-oo
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\left(x - 1\right) \cot{\left(\pi \left(x - 1\right) \right)}\right) = \frac{1}{\pi}$$
$$\lim_{x \to \infty}\left(\left(x - 1\right) \cot{\left(\pi \left(x - 1\right) \right)}\right)$$
More at x→oo
$$\lim_{x \to 0^-}\left(\left(x - 1\right) \cot{\left(\pi \left(x - 1\right) \right)}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\left(x - 1\right) \cot{\left(\pi \left(x - 1\right) \right)}\right) = -\infty$$
More at x→0 from the right
$$\lim_{x \to -\infty}\left(\left(x - 1\right) \cot{\left(\pi \left(x - 1\right) \right)}\right)$$
More at x→-oo