Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of x^2*(1/3-cos(8*x)/3)
-
Limit of 7^(1/(-3+x))
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Limit of (-6-x^2-3*x+4*x^3)/(3-x^2+2*x^3)
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Limit of ((-4+3*x)/(2+3*x))^(1+x)/3
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Identical expressions
- (- four +x^ two)/(three -sqrt(seven +x))
- ( minus 4 plus x squared ) divide by (3 minus square root of (7 plus x))
- ( minus four plus x to the power of two) divide by (three minus square root of (seven plus x))
- (-4+x^2)/(3-√(7+x))
- (-4+x2)/(3-sqrt(7+x))
- -4+x2/3-sqrt7+x
- (-4+x²)/(3-sqrt(7+x))
- (-4+x to the power of 2)/(3-sqrt(7+x))
- -4+x^2/3-sqrt7+x
- (-4+x^2) divide by (3-sqrt(7+x))
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Similar expressions
- (-4+x^2)/(3-sqrt(7-x))
- (-4+x^2)/(3+sqrt(7+x))
- (-4-x^2)/(3-sqrt(7+x))
- (4+x^2)/(3-sqrt(7+x))
Limit of the function (-4+x^2)/(3-sqrt(7+x))
The solution
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[src]
/ 2 \
| -4 + x |
lim |-------------|
x->2+| _______|
\3 - \/ 7 + x /
$$\lim_{x \to 2^+}\left(\frac{x^{2} - 4}{3 - \sqrt{x + 7}}\right)$$
Limit((-4 + x^2)/(3 - sqrt(7 + x)), x, 2)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 2^+}\left(x^{2} - 4\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 2^+}\left(3 - \sqrt{x + 7}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 2^+}\left(\frac{x^{2} - 4}{3 - \sqrt{x + 7}}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{\frac{d}{d x} \left(x^{2} - 4\right)}{\frac{d}{d x} \left(3 - \sqrt{x + 7}\right)}\right)$$
=
$$\lim_{x \to 2^+}\left(- 4 x \sqrt{x + 7}\right)$$
=
$$\lim_{x \to 2^+} -24$$
=
$$\lim_{x \to 2^+} -24$$
=
$$-24$$
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
$$\lim_{x \to 2^+}\left(x^{2} - 4\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 2^+}\left(3 - \sqrt{x + 7}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 2^+}\left(\frac{x^{2} - 4}{3 - \sqrt{x + 7}}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{\frac{d}{d x} \left(x^{2} - 4\right)}{\frac{d}{d x} \left(3 - \sqrt{x + 7}\right)}\right)$$
=
$$\lim_{x \to 2^+}\left(- 4 x \sqrt{x + 7}\right)$$
=
$$\lim_{x \to 2^+} -24$$
=
$$\lim_{x \to 2^+} -24$$
=
$$-24$$
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]
/ 2 \
| -4 + x |
lim |-------------|
x->2+| _______|
\3 - \/ 7 + x /
$$\lim_{x \to 2^+}\left(\frac{x^{2} - 4}{3 - \sqrt{x + 7}}\right)$$
-24
$$-24$$
= -24.0
/ 2 \
| -4 + x |
lim |-------------|
x->2-| _______|
\3 - \/ 7 + x /
$$\lim_{x \to 2^-}\left(\frac{x^{2} - 4}{3 - \sqrt{x + 7}}\right)$$
-24
$$-24$$
= -24.0
= -24.0
The graph