Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of (-3+2*log((3+x)/x))/x
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Limit of (15-28*x^5-19*x+16*x^2+19*x^3)/(15-30*x^2-7*x^5+11*x^4+23*x)
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Limit of (6-30*x^5-24*x^6-15*x-8*x^2+15*x^4)/(22-7*x^2+5*x^6+10*x^4+26*x^3+26*x^5+27*x)
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Limit of 2+(-1)^x
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Identical expressions
- five *x/atan(seven *x)
- 5 multiply by x divide by arc tangent of gent of (7 multiply by x)
- five multiply by x divide by arc tangent of gent of (seven multiply by x)
- 5x/atan(7x)
- 5x/atan7x
- 5*x divide by atan(7*x)
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Similar expressions
- sin(5*x)/atan(7*x)
- 5*x/arctan(7*x)
Limit of the function 5*x/atan(7*x)
The solution
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[src]
/ 5*x \ lim |---------| x->0+\atan(7*x)/
$$\lim_{x \to 0^+}\left(\frac{5 x}{\operatorname{atan}{\left(7 x \right)}}\right)$$
Limit((5*x)/atan(7*x), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(5 x\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \operatorname{atan}{\left(7 x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{5 x}{\operatorname{atan}{\left(7 x \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{5 x}{\operatorname{atan}{\left(7 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 5 x}{\frac{d}{d x} \operatorname{atan}{\left(7 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(35 x^{2} + \frac{5}{7}\right)$$
=
$$\lim_{x \to 0^+} \frac{5}{7}$$
=
$$\lim_{x \to 0^+} \frac{5}{7}$$
=
$$\frac{5}{7}$$
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
$$\lim_{x \to 0^+}\left(5 x\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \operatorname{atan}{\left(7 x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{5 x}{\operatorname{atan}{\left(7 x \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{5 x}{\operatorname{atan}{\left(7 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} 5 x}{\frac{d}{d x} \operatorname{atan}{\left(7 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(35 x^{2} + \frac{5}{7}\right)$$
=
$$\lim_{x \to 0^+} \frac{5}{7}$$
=
$$\lim_{x \to 0^+} \frac{5}{7}$$
=
$$\frac{5}{7}$$
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]
/ 5*x \ lim |---------| x->0+\atan(7*x)/
$$\lim_{x \to 0^+}\left(\frac{5 x}{\operatorname{atan}{\left(7 x \right)}}\right)$$
5/7
$$\frac{5}{7}$$
= 0.714285714285714
/ 5*x \ lim |---------| x->0-\atan(7*x)/
$$\lim_{x \to 0^-}\left(\frac{5 x}{\operatorname{atan}{\left(7 x \right)}}\right)$$
5/7
$$\frac{5}{7}$$
= 0.714285714285714
= 0.714285714285714
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{5 x}{\operatorname{atan}{\left(7 x \right)}}\right) = \frac{5}{7}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{5 x}{\operatorname{atan}{\left(7 x \right)}}\right) = \frac{5}{7}$$
$$\lim_{x \to \infty}\left(\frac{5 x}{\operatorname{atan}{\left(7 x \right)}}\right) = \infty$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{5 x}{\operatorname{atan}{\left(7 x \right)}}\right) = \frac{5}{\operatorname{atan}{\left(7 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{5 x}{\operatorname{atan}{\left(7 x \right)}}\right) = \frac{5}{\operatorname{atan}{\left(7 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{5 x}{\operatorname{atan}{\left(7 x \right)}}\right) = \infty$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{5 x}{\operatorname{atan}{\left(7 x \right)}}\right) = \frac{5}{7}$$
$$\lim_{x \to \infty}\left(\frac{5 x}{\operatorname{atan}{\left(7 x \right)}}\right) = \infty$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{5 x}{\operatorname{atan}{\left(7 x \right)}}\right) = \frac{5}{\operatorname{atan}{\left(7 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{5 x}{\operatorname{atan}{\left(7 x \right)}}\right) = \frac{5}{\operatorname{atan}{\left(7 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{5 x}{\operatorname{atan}{\left(7 x \right)}}\right) = \infty$$
More at x→-oo
The graph