Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (1+x^2+9*x)/(-5+2*x+7*x^2)
-
Limit of ((5+x^2-6*x)/(5+x^2-5*x))^(2+3*x)
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Limit of (2+x^2+3*x)/(-4+x^2)
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Limit of (-8+x^2+2*x)/(-2+x)
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Identical expressions
- tan(pi*x)/(- three +x)
- tangent of ( Pi multiply by x) divide by ( minus 3 plus x)
- tangent of ( Pi multiply by x) divide by ( minus three plus x)
- tan(pix)/(-3+x)
- tanpix/-3+x
- tan(pi*x) divide by (-3+x)
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Similar expressions
- tan(pi*x)/(-3-x)
- tan(pi*x)/(3+x)
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What you mean?
- tan(pi*x)/(-3 + x)
- tan(pi*x)/(-3) + x
- tan(pi*x)/(-3 + x)
- tan(pi*x)/(-3) + x
- tan(pi*x)/(-3 + x)
- tan(pi*x)/(-3) + x
- tan(pi*x)/(-3) + x
You entered:
tan(pi*x)/(-3+x)
What you mean?
Limit of the function tan(pi*x)/(-3+x)
The solution
You have entered
[src]
/tan(pi*x)\ lim |---------| x->3+\ -3 + x /
$$\lim_{x \to 3^+}\left(\frac{\tan{\left(\pi x \right)}}{x - 3}\right)$$
Limit(tan(pi*x)/(-3 + x), x, 3)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 3^+} \tan{\left(\pi x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 3^+}\left(x - 3\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 3^+}\left(\frac{\tan{\left(\pi x \right)}}{x - 3}\right)$$
=
$$\lim_{x \to 3^+}\left(\frac{\frac{d}{d x} \tan{\left(\pi x \right)}}{\frac{d}{d x} \left(x - 3\right)}\right)$$
=
$$\lim_{x \to 3^+}\left(\pi \left(\tan^{2}{\left(\pi x \right)} + 1\right)\right)$$
=
$$\lim_{x \to 3^+} \pi$$
=
$$\lim_{x \to 3^+} \pi$$
=
$$\pi$$
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
$$\lim_{x \to 3^+} \tan{\left(\pi x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 3^+}\left(x - 3\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 3^+}\left(\frac{\tan{\left(\pi x \right)}}{x - 3}\right)$$
=
$$\lim_{x \to 3^+}\left(\frac{\frac{d}{d x} \tan{\left(\pi x \right)}}{\frac{d}{d x} \left(x - 3\right)}\right)$$
=
$$\lim_{x \to 3^+}\left(\pi \left(\tan^{2}{\left(\pi x \right)} + 1\right)\right)$$
=
$$\lim_{x \to 3^+} \pi$$
=
$$\lim_{x \to 3^+} \pi$$
=
$$\pi$$
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(pi*x)\ lim |---------| x->3+\ -3 + x /
$$\lim_{x \to 3^+}\left(\frac{\tan{\left(\pi x \right)}}{x - 3}\right)$$
pi
$$\pi$$
= 3.16744494160142
/tan(pi*x)\ lim |---------| x->3-\ -3 + x /
$$\lim_{x \to 3^-}\left(\frac{\tan{\left(\pi x \right)}}{x - 3}\right)$$
pi
$$\pi$$
= 3.16744494160142
= 3.16744494160142
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 3^-}\left(\frac{\tan{\left(\pi x \right)}}{x - 3}\right) = \pi$$
More at x→3 from the left
$$\lim_{x \to 3^+}\left(\frac{\tan{\left(\pi x \right)}}{x - 3}\right) = \pi$$
$$\lim_{x \to \infty}\left(\frac{\tan{\left(\pi x \right)}}{x - 3}\right)$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{\tan{\left(\pi x \right)}}{x - 3}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(\pi x \right)}}{x - 3}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{\tan{\left(\pi x \right)}}{x - 3}\right) = 0$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\tan{\left(\pi x \right)}}{x - 3}\right) = 0$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(\pi x \right)}}{x - 3}\right)$$
More at x→-oo
More at x→3 from the left
$$\lim_{x \to 3^+}\left(\frac{\tan{\left(\pi x \right)}}{x - 3}\right) = \pi$$
$$\lim_{x \to \infty}\left(\frac{\tan{\left(\pi x \right)}}{x - 3}\right)$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{\tan{\left(\pi x \right)}}{x - 3}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\tan{\left(\pi x \right)}}{x - 3}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{\tan{\left(\pi x \right)}}{x - 3}\right) = 0$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\tan{\left(\pi x \right)}}{x - 3}\right) = 0$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(\pi x \right)}}{x - 3}\right)$$
More at x→-oo
The graph