Mister Exam
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How to use it?
- Limit of the function:
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Limit of (9+3*x^2+4*x)/(7-7*x+3*x^2)
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Limit of -pi+(1+cos(x))/x
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Limit of (1-5*x)^(1/x)
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Limit of (1+2/n)^(4*n)
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Identical expressions
- -pi+(one +cos(x))/x
- minus Pi plus (1 plus co sinus of e of (x)) divide by x
- minus Pi plus (one plus co sinus of e of (x)) divide by x
- -pi+1+cosx/x
- -pi+(1+cos(x)) divide by x
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Similar expressions
- pi+(1+cos(x))/x
- -pi-(1+cos(x))/x
- -pi+(1-cos(x))/x
- -pi+(1+cosx)/x
Limit of the function -pi+(1+cos(x))/x
The solution
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/ 1 + cos(x)\ lim |-pi + ----------| x->pi+\ x /
$$\lim_{x \to \pi^+}\left(- \pi + \frac{\cos{\left(x \right)} + 1}{x}\right)$$
Limit(-pi + (1 + cos(x))/x, x, pi)
Lopital's rule
There is no sense to apply Lopital's rule to this function since there is no indeterminateness of 0/0 or oo/oo type
The graph
One‐sided limits
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/ 1 + cos(x)\ lim |-pi + ----------| x->pi+\ x /
$$\lim_{x \to \pi^+}\left(- \pi + \frac{\cos{\left(x \right)} + 1}{x}\right)$$
-pi
$$- \pi$$
= -3.14159265358979
/ 1 + cos(x)\ lim |-pi + ----------| x->pi-\ x /
$$\lim_{x \to \pi^-}\left(- \pi + \frac{\cos{\left(x \right)} + 1}{x}\right)$$
-pi
$$- \pi$$
= -3.14159265358979
= -3.14159265358979
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \pi^-}\left(- \pi + \frac{\cos{\left(x \right)} + 1}{x}\right) = - \pi$$
More at x→pi from the left
$$\lim_{x \to \pi^+}\left(- \pi + \frac{\cos{\left(x \right)} + 1}{x}\right) = - \pi$$
$$\lim_{x \to \infty}\left(- \pi + \frac{\cos{\left(x \right)} + 1}{x}\right) = - \pi$$
More at x→oo
$$\lim_{x \to 0^-}\left(- \pi + \frac{\cos{\left(x \right)} + 1}{x}\right) = -\infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(- \pi + \frac{\cos{\left(x \right)} + 1}{x}\right) = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(- \pi + \frac{\cos{\left(x \right)} + 1}{x}\right) = - \pi + \cos{\left(1 \right)} + 1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(- \pi + \frac{\cos{\left(x \right)} + 1}{x}\right) = - \pi + \cos{\left(1 \right)} + 1$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(- \pi + \frac{\cos{\left(x \right)} + 1}{x}\right) = - \pi$$
More at x→-oo
More at x→pi from the left
$$\lim_{x \to \pi^+}\left(- \pi + \frac{\cos{\left(x \right)} + 1}{x}\right) = - \pi$$
$$\lim_{x \to \infty}\left(- \pi + \frac{\cos{\left(x \right)} + 1}{x}\right) = - \pi$$
More at x→oo
$$\lim_{x \to 0^-}\left(- \pi + \frac{\cos{\left(x \right)} + 1}{x}\right) = -\infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(- \pi + \frac{\cos{\left(x \right)} + 1}{x}\right) = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(- \pi + \frac{\cos{\left(x \right)} + 1}{x}\right) = - \pi + \cos{\left(1 \right)} + 1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(- \pi + \frac{\cos{\left(x \right)} + 1}{x}\right) = - \pi + \cos{\left(1 \right)} + 1$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(- \pi + \frac{\cos{\left(x \right)} + 1}{x}\right) = - \pi$$
More at x→-oo
The graph