Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of x^2*(1/3-cos(8*x)/3)
-
Limit of (-2+x^2-x)/(-2+x+3*x^2)
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Limit of (-1+sin(x))/cos(x)
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Limit of (-2*sin(x)+sin(2*x))/(x*log(cos(5*x)))
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Identical expressions
- (- one +sin(x))/cos(x)
- ( minus 1 plus sinus of (x)) divide by co sinus of e of (x)
- ( minus one plus sinus of (x)) divide by co sinus of e of (x)
- -1+sinx/cosx
- (-1+sin(x)) divide by cos(x)
-
Similar expressions
- (1+sin(x))/cos(x)
- (-1-sin(x))/cos(x)
- (-1+sinx)/cosx
Limit of the function (-1+sin(x))/cos(x)
The solution
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/-1 + sin(x)\ lim |-----------| pi \ cos(x) / x->--+ 2
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\sin{\left(x \right)} - 1}{\cos{\left(x \right)}}\right)$$
Limit((-1 + sin(x))/cos(x), x, pi/2)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \frac{\pi}{2}^+}\left(\sin{\left(x \right)} - 1\right) = 0$$
and limit for the denominator is
$$\lim_{x \to \frac{\pi}{2}^+} \cos{\left(x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\sin{\left(x \right)} - 1}{\cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \left(\sin{\left(x \right)} - 1\right)}{\frac{d}{d x} \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(- \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(- \cos{\left(x \right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(- \cos{\left(x \right)}\right)$$
=
$$0$$
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
$$\lim_{x \to \frac{\pi}{2}^+}\left(\sin{\left(x \right)} - 1\right) = 0$$
and limit for the denominator is
$$\lim_{x \to \frac{\pi}{2}^+} \cos{\left(x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\sin{\left(x \right)} - 1}{\cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \left(\sin{\left(x \right)} - 1\right)}{\frac{d}{d x} \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(- \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(- \cos{\left(x \right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(- \cos{\left(x \right)}\right)$$
=
$$0$$
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]
/-1 + sin(x)\ lim |-----------| pi \ cos(x) / x->--+ 2
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\sin{\left(x \right)} - 1}{\cos{\left(x \right)}}\right)$$
0
$$0$$
= -3.06161699786838e-17
/-1 + sin(x)\ lim |-----------| pi \ cos(x) / x->--- 2
$$\lim_{x \to \frac{\pi}{2}^-}\left(\frac{\sin{\left(x \right)} - 1}{\cos{\left(x \right)}}\right)$$
0
$$0$$
= -3.06161699786839e-17
= -3.06161699786839e-17
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \frac{\pi}{2}^-}\left(\frac{\sin{\left(x \right)} - 1}{\cos{\left(x \right)}}\right) = 0$$
More at x→pi/2 from the left
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\sin{\left(x \right)} - 1}{\cos{\left(x \right)}}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} - 1}{\cos{\left(x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)} - 1}{\cos{\left(x \right)}}\right) = -1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} - 1}{\cos{\left(x \right)}}\right) = -1$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(x \right)} - 1}{\cos{\left(x \right)}}\right) = \frac{-1 + \sin{\left(1 \right)}}{\cos{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(x \right)} - 1}{\cos{\left(x \right)}}\right) = \frac{-1 + \sin{\left(1 \right)}}{\cos{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} - 1}{\cos{\left(x \right)}}\right)$$
More at x→-oo
More at x→pi/2 from the left
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\sin{\left(x \right)} - 1}{\cos{\left(x \right)}}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} - 1}{\cos{\left(x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)} - 1}{\cos{\left(x \right)}}\right) = -1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} - 1}{\cos{\left(x \right)}}\right) = -1$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(x \right)} - 1}{\cos{\left(x \right)}}\right) = \frac{-1 + \sin{\left(1 \right)}}{\cos{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(x \right)} - 1}{\cos{\left(x \right)}}\right) = \frac{-1 + \sin{\left(1 \right)}}{\cos{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} - 1}{\cos{\left(x \right)}}\right)$$
More at x→-oo
The graph