Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of (1-log(7*x))^(7*x)
-
Limit of sin(x)/sqrt(1-cos(x))
-
Limit of (e^x-e^2)/(-2+x)
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Limit of tan(3*x)/(7*x)
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Identical expressions
- tan(three *x)/(seven *x)
- tangent of (3 multiply by x) divide by (7 multiply by x)
- tangent of (three multiply by x) divide by (seven multiply by x)
- tan(3x)/(7x)
- tan3x/7x
- tan(3*x) divide by (7*x)
Limit of the function tan(3*x)/(7*x)
The solution
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/tan(3*x)\ lim |--------| x->0+\ 7*x /
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right)$$
Limit(tan(3*x)/((7*x)), x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right)$$
transform
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right) = \lim_{x \to 0^+}\left(\frac{\frac{1}{7 x} \sin{\left(3 x \right)}}{\cos{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{7 x} \sin{\left(3 x \right)}\right) \lim_{x \to 0^+} \frac{1}{\cos{\left(3 x \right)}} = \lim_{x \to 0^+}\left(\frac{1}{7 x} \sin{\left(3 x \right)}\right)$$
Do replacement
$$u = 3 x$$
then
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right) = \lim_{u \to 0^+}\left(\frac{3 \sin{\left(u \right)}}{7 u}\right)$$
=
$$\frac{3 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)}{7}$$
The limit
$$\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
is first remarkable limit, is equal to 1.
The final answer:
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right) = \frac{3}{7}$$
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right)$$
transform
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right) = \lim_{x \to 0^+}\left(\frac{\frac{1}{7 x} \sin{\left(3 x \right)}}{\cos{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{7 x} \sin{\left(3 x \right)}\right) \lim_{x \to 0^+} \frac{1}{\cos{\left(3 x \right)}} = \lim_{x \to 0^+}\left(\frac{1}{7 x} \sin{\left(3 x \right)}\right)$$
Do replacement
$$u = 3 x$$
then
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{7 x}\right) = \lim_{u \to 0^+}\left(\frac{3 \sin{\left(u \right)}}{7 u}\right)$$
=
$$\frac{3 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)}{7}$$
The limit
$$\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
is first remarkable limit, is equal to 1.
The final answer:
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right) = \frac{3}{7}$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \tan{\left(3 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(7 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tan{\left(3 x \right)}}{\frac{d}{d x} 7 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{3 \tan^{2}{\left(3 x \right)}}{7} + \frac{3}{7}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{3 \tan^{2}{\left(3 x \right)}}{7} + \frac{3}{7}\right)$$
=
$$\frac{3}{7}$$
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
$$\lim_{x \to 0^+} \tan{\left(3 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(7 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tan{\left(3 x \right)}}{\frac{d}{d x} 7 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{3 \tan^{2}{\left(3 x \right)}}{7} + \frac{3}{7}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{3 \tan^{2}{\left(3 x \right)}}{7} + \frac{3}{7}\right)$$
=
$$\frac{3}{7}$$
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]
/tan(3*x)\ lim |--------| x->0+\ 7*x /
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right)$$
3/7
$$\frac{3}{7}$$
= 0.428571428571429
/tan(3*x)\ lim |--------| x->0-\ 7*x /
$$\lim_{x \to 0^-}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right)$$
3/7
$$\frac{3}{7}$$
= 0.428571428571429
= 0.428571428571429
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right) = \frac{3}{7}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right) = \frac{3}{7}$$
$$\lim_{x \to \infty}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right) = \frac{\tan{\left(3 \right)}}{7}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right) = \frac{\tan{\left(3 \right)}}{7}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right) = \frac{3}{7}$$
$$\lim_{x \to \infty}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right) = \frac{\tan{\left(3 \right)}}{7}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right) = \frac{\tan{\left(3 \right)}}{7}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(3 x \right)}}{7 x}\right)$$
More at x→-oo
The graph