Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of 2^(-n)*2^(1+n)
- Limit of tan(k*x)/x
-
Limit of (-x+tan(x))/(x+2*sin(x))
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Limit of asin(5*x)/tan(3*x)
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Identical expressions
- tan(k*x)/x
- tangent of (k multiply by x) divide by x
- tan(kx)/x
- tankx/x
- tan(k*x) divide by x
Limit of the function tan(k*x)/x
The solution
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/tan(k*x)\ lim |--------| x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(k x \right)}}{x}\right)$$
Limit(tan(k*x)/x, x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(k x \right)}}{x}\right)$$
transform
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(k x \right)}}{x}\right) = \lim_{x \to 0^+}\left(\frac{\sin{\left(k x \right)}}{x \cos{\left(k x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(k x \right)}}{x}\right) \lim_{x \to 0^+} \frac{1}{\cos{\left(k x \right)}} = \lim_{x \to 0^+}\left(\frac{\sin{\left(k x \right)}}{x}\right)$$
Do replacement
$$u = k x$$
then
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(k x \right)}}{x}\right) = \lim_{u \to 0^+}\left(\frac{k \sin{\left(u \right)}}{u}\right)$$
=
$$k \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
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:
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(k x \right)}}{x}\right) = k$$
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(k x \right)}}{x}\right)$$
transform
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(k x \right)}}{x}\right) = \lim_{x \to 0^+}\left(\frac{\sin{\left(k x \right)}}{x \cos{\left(k x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(k x \right)}}{x}\right) \lim_{x \to 0^+} \frac{1}{\cos{\left(k x \right)}} = \lim_{x \to 0^+}\left(\frac{\sin{\left(k x \right)}}{x}\right)$$
Do replacement
$$u = k x$$
then
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(k x \right)}}{x}\right) = \lim_{u \to 0^+}\left(\frac{k \sin{\left(u \right)}}{u}\right)$$
=
$$k \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
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:
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(k x \right)}}{x}\right) = k$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \tan{\left(k x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(k x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(k x \right)}}{x}\right)$$
=
$$k$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 0 time(s)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \tan{\left(k x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(k x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(k x \right)}}{x}\right)$$
=
$$k$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 0 time(s)
One‐sided limits
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/tan(k*x)\ lim |--------| x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(k x \right)}}{x}\right)$$
k
$$k$$
/tan(k*x)\ lim |--------| x->0-\ x /
$$\lim_{x \to 0^-}\left(\frac{\tan{\left(k x \right)}}{x}\right)$$
k
$$k$$
k
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\tan{\left(k x \right)}}{x}\right) = k$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(k x \right)}}{x}\right) = k$$
$$\lim_{x \to \infty}\left(\frac{\tan{\left(k x \right)}}{x}\right) = \tilde{\infty} k \tan^{2}{\left(\tilde{\infty} k \right)} + \tilde{\infty} k$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\tan{\left(k x \right)}}{x}\right) = \tan{\left(k \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\tan{\left(k x \right)}}{x}\right) = \tan{\left(k \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(k x \right)}}{x}\right) = \tilde{\infty} k \tan^{2}{\left(\tilde{\infty} k \right)} + \tilde{\infty} k$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(k x \right)}}{x}\right) = k$$
$$\lim_{x \to \infty}\left(\frac{\tan{\left(k x \right)}}{x}\right) = \tilde{\infty} k \tan^{2}{\left(\tilde{\infty} k \right)} + \tilde{\infty} k$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\tan{\left(k x \right)}}{x}\right) = \tan{\left(k \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\tan{\left(k x \right)}}{x}\right) = \tan{\left(k \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(k x \right)}}{x}\right) = \tilde{\infty} k \tan^{2}{\left(\tilde{\infty} k \right)} + \tilde{\infty} k$$
More at x→-oo