Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of (-2*x^2+4*x^3+5*x)/(3*x^2+7*x)
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Limit of (1-3*x^2+2*x^3)/(x^3+2*x+4*x^2)
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Limit of 5+3*n
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Limit of (1-cos(4*x))/(5*x)
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Identical expressions
- (one -cos(four *x))/(five *x)
- (1 minus co sinus of e of (4 multiply by x)) divide by (5 multiply by x)
- (one minus co sinus of e of (four multiply by x)) divide by (five multiply by x)
- (1-cos(4x))/(5x)
- 1-cos4x/5x
- (1-cos(4*x)) divide by (5*x)
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Similar expressions
- (1+cos(4*x))/(5*x)
Limit of the function (1-cos(4*x))/(5*x)
The solution
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[src]
/1 - cos(4*x)\ lim |------------| x->0+\ 5*x /
$$\lim_{x \to 0^+}\left(\frac{1 - \cos{\left(4 x \right)}}{5 x}\right)$$
Limit((1 - cos(4*x))/((5*x)), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(1 - \cos{\left(4 x \right)}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(5 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{1 - \cos{\left(4 x \right)}}{5 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{1 - \cos{\left(4 x \right)}}{5 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(1 - \cos{\left(4 x \right)}\right)}{\frac{d}{d x} 5 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{4 \sin{\left(4 x \right)}}{5}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{4 \sin{\left(4 x \right)}}{5}\right)$$
=
$$0$$
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
$$\lim_{x \to 0^+}\left(1 - \cos{\left(4 x \right)}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(5 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{1 - \cos{\left(4 x \right)}}{5 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{1 - \cos{\left(4 x \right)}}{5 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(1 - \cos{\left(4 x \right)}\right)}{\frac{d}{d x} 5 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{4 \sin{\left(4 x \right)}}{5}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{4 \sin{\left(4 x \right)}}{5}\right)$$
=
$$0$$
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 - cos(4*x)\ lim |------------| x->0+\ 5*x /
$$\lim_{x \to 0^+}\left(\frac{1 - \cos{\left(4 x \right)}}{5 x}\right)$$
0
$$0$$
= 7.85254963568473e-31
/1 - cos(4*x)\ lim |------------| x->0-\ 5*x /
$$\lim_{x \to 0^-}\left(\frac{1 - \cos{\left(4 x \right)}}{5 x}\right)$$
0
$$0$$
= -7.85254963568473e-31
= -7.85254963568473e-31
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{1 - \cos{\left(4 x \right)}}{5 x}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{1 - \cos{\left(4 x \right)}}{5 x}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{1 - \cos{\left(4 x \right)}}{5 x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{1 - \cos{\left(4 x \right)}}{5 x}\right) = \frac{1}{5} - \frac{\cos{\left(4 \right)}}{5}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{1 - \cos{\left(4 x \right)}}{5 x}\right) = \frac{1}{5} - \frac{\cos{\left(4 \right)}}{5}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{1 - \cos{\left(4 x \right)}}{5 x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{1 - \cos{\left(4 x \right)}}{5 x}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{1 - \cos{\left(4 x \right)}}{5 x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{1 - \cos{\left(4 x \right)}}{5 x}\right) = \frac{1}{5} - \frac{\cos{\left(4 \right)}}{5}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{1 - \cos{\left(4 x \right)}}{5 x}\right) = \frac{1}{5} - \frac{\cos{\left(4 \right)}}{5}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{1 - \cos{\left(4 x \right)}}{5 x}\right) = 0$$
More at x→-oo
The graph