Mister Exam
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How to use it?
- Limit of the function:
-
Limit of (2*x/(-3+2*x))^(3*x)
- Limit of (1-cos(a*x))/(1-cos(b*x))
-
Limit of (1-3/x)^x
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Limit of 5^x
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Identical expressions
- (one -cos(a*x))/(one -cos(b*x))
- (1 minus co sinus of e of (a multiply by x)) divide by (1 minus co sinus of e of (b multiply by x))
- (one minus co sinus of e of (a multiply by x)) divide by (one minus co sinus of e of (b multiply by x))
- (1-cos(ax))/(1-cos(bx))
- 1-cosax/1-cosbx
- (1-cos(a*x)) divide by (1-cos(b*x))
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Similar expressions
- (1-cos(a*x))/(1+cos(b*x))
- 1-cos(a*x)/(1-cos(b*x))
- (1+cos(a*x))/(1-cos(b*x))
Limit of the function (1-cos(a*x))/(1-cos(b*x))
The solution
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/1 - cos(a*x)\ lim |------------| x->0+\1 - cos(b*x)/
$$\lim_{x \to 0^+}\left(\frac{- \cos{\left(a x \right)} + 1}{- \cos{\left(b x \right)} + 1}\right)$$
Limit((1 - cos(a*x))/(1 - cos(b*x)), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(- \cos{\left(a x \right)} + 1\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(- \cos{\left(b x \right)} + 1\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{- \cos{\left(a x \right)} + 1}{- \cos{\left(b x \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{\partial}{\partial x} \left(- \cos{\left(a x \right)} + 1\right)}{\frac{\partial}{\partial x} \left(- \cos{\left(b x \right)} + 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{a \sin{\left(a x \right)}}{b \sin{\left(b x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{\partial}{\partial x} a \sin{\left(a x \right)}}{\frac{\partial}{\partial x} b \sin{\left(b x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{a^{2} \cos{\left(a x \right)}}{b^{2} \cos{\left(b x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{a^{2}}{b^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{a^{2}}{b^{2}}\right)$$
=
$$\frac{a^{2}}{b^{2}}$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(- \cos{\left(a x \right)} + 1\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(- \cos{\left(b x \right)} + 1\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{- \cos{\left(a x \right)} + 1}{- \cos{\left(b x \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{\partial}{\partial x} \left(- \cos{\left(a x \right)} + 1\right)}{\frac{\partial}{\partial x} \left(- \cos{\left(b x \right)} + 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{a \sin{\left(a x \right)}}{b \sin{\left(b x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{\partial}{\partial x} a \sin{\left(a x \right)}}{\frac{\partial}{\partial x} b \sin{\left(b x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{a^{2} \cos{\left(a x \right)}}{b^{2} \cos{\left(b x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{a^{2}}{b^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{a^{2}}{b^{2}}\right)$$
=
$$\frac{a^{2}}{b^{2}}$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)
One‐sided limits
[src]
/1 - cos(a*x)\ lim |------------| x->0+\1 - cos(b*x)/
$$\lim_{x \to 0^+}\left(\frac{- \cos{\left(a x \right)} + 1}{- \cos{\left(b x \right)} + 1}\right)$$
2 a -- 2 b
$$\frac{a^{2}}{b^{2}}$$
/1 - cos(a*x)\ lim |------------| x->0-\1 - cos(b*x)/
$$\lim_{x \to 0^-}\left(\frac{- \cos{\left(a x \right)} + 1}{- \cos{\left(b x \right)} + 1}\right)$$
2 a -- 2 b
$$\frac{a^{2}}{b^{2}}$$
a^2/b^2
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{- \cos{\left(a x \right)} + 1}{- \cos{\left(b x \right)} + 1}\right) = \frac{a^{2}}{b^{2}}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{- \cos{\left(a x \right)} + 1}{- \cos{\left(b x \right)} + 1}\right) = \frac{a^{2}}{b^{2}}$$
$$\lim_{x \to \infty}\left(\frac{- \cos{\left(a x \right)} + 1}{- \cos{\left(b x \right)} + 1}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{- \cos{\left(a x \right)} + 1}{- \cos{\left(b x \right)} + 1}\right) = \frac{\cos{\left(a \right)} - 1}{\cos{\left(b \right)} - 1}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{- \cos{\left(a x \right)} + 1}{- \cos{\left(b x \right)} + 1}\right) = \frac{\cos{\left(a \right)} - 1}{\cos{\left(b \right)} - 1}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{- \cos{\left(a x \right)} + 1}{- \cos{\left(b x \right)} + 1}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{- \cos{\left(a x \right)} + 1}{- \cos{\left(b x \right)} + 1}\right) = \frac{a^{2}}{b^{2}}$$
$$\lim_{x \to \infty}\left(\frac{- \cos{\left(a x \right)} + 1}{- \cos{\left(b x \right)} + 1}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{- \cos{\left(a x \right)} + 1}{- \cos{\left(b x \right)} + 1}\right) = \frac{\cos{\left(a \right)} - 1}{\cos{\left(b \right)} - 1}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{- \cos{\left(a x \right)} + 1}{- \cos{\left(b x \right)} + 1}\right) = \frac{\cos{\left(a \right)} - 1}{\cos{\left(b \right)} - 1}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{- \cos{\left(a x \right)} + 1}{- \cos{\left(b x \right)} + 1}\right)$$
More at x→-oo