Mister Exam
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How to use it?
- Limit of the function:
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Limit of (-5+7*n)/(2+n)
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Limit of (1+6*x)^(1/x)
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Limit of (-2+x)/(-4+x)
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Limit of (-16+x^2-6*x)/(-2+x+x^2)
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Identical expressions
- (- one - two *x+ three *x^ two)/(one +x^ two + four *x)
- ( minus 1 minus 2 multiply by x plus 3 multiply by x squared ) divide by (1 plus x squared plus 4 multiply by x)
- ( minus one minus two multiply by x plus three multiply by x to the power of two) divide by (one plus x to the power of two plus four multiply by x)
- (-1-2*x+3*x2)/(1+x2+4*x)
- -1-2*x+3*x2/1+x2+4*x
- (-1-2*x+3*x²)/(1+x²+4*x)
- (-1-2*x+3*x to the power of 2)/(1+x to the power of 2+4*x)
- (-1-2x+3x^2)/(1+x^2+4x)
- (-1-2x+3x2)/(1+x2+4x)
- -1-2x+3x2/1+x2+4x
- -1-2x+3x^2/1+x^2+4x
- (-1-2*x+3*x^2) divide by (1+x^2+4*x)
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Similar expressions
- (-1-2*x+3*x^2)/(1-x^2+4*x)
- (-1-2*x-3*x^2)/(1+x^2+4*x)
- (-1+2*x+3*x^2)/(1+x^2+4*x)
- (1-2*x+3*x^2)/(1+x^2+4*x)
- (-1-2*x+3*x^2)/(1+x^2-4*x)
Limit of the function (-1-2*x+3*x^2)/(1+x^2+4*x)
The solution
You have entered
[src]
/ 2\
|-1 - 2*x + 3*x |
lim |---------------|
x->1+| 2 |
\ 1 + x + 4*x /
$$\lim_{x \to 1^+}\left(\frac{3 x^{2} + \left(- 2 x - 1\right)}{4 x + \left(x^{2} + 1\right)}\right)$$
Limit((-1 - 2*x + 3*x^2)/(1 + x^2 + 4*x), x, 1)
Detail solution
Let's take the limit
$$\lim_{x \to 1^+}\left(\frac{3 x^{2} + \left(- 2 x - 1\right)}{4 x + \left(x^{2} + 1\right)}\right)$$
transform
$$\lim_{x \to 1^+}\left(\frac{3 x^{2} + \left(- 2 x - 1\right)}{4 x + \left(x^{2} + 1\right)}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\left(x - 1\right) \left(3 x + 1\right)}{x^{2} + 4 x + 1}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\left(x - 1\right) \left(3 x + 1\right)}{x^{2} + 4 x + 1}\right) = $$
$$\frac{\left(-1 + 1\right) \left(1 + 3\right)}{1 + 1^{2} + 4} = $$
The final answer:
$$\lim_{x \to 1^+}\left(\frac{3 x^{2} + \left(- 2 x - 1\right)}{4 x + \left(x^{2} + 1\right)}\right) = 0$$
$$\lim_{x \to 1^+}\left(\frac{3 x^{2} + \left(- 2 x - 1\right)}{4 x + \left(x^{2} + 1\right)}\right)$$
transform
$$\lim_{x \to 1^+}\left(\frac{3 x^{2} + \left(- 2 x - 1\right)}{4 x + \left(x^{2} + 1\right)}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\left(x - 1\right) \left(3 x + 1\right)}{x^{2} + 4 x + 1}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\left(x - 1\right) \left(3 x + 1\right)}{x^{2} + 4 x + 1}\right) = $$
$$\frac{\left(-1 + 1\right) \left(1 + 3\right)}{1 + 1^{2} + 4} = $$
= 0
The final answer:
$$\lim_{x \to 1^+}\left(\frac{3 x^{2} + \left(- 2 x - 1\right)}{4 x + \left(x^{2} + 1\right)}\right) = 0$$
Lopital's rule
There is no sense to apply Lopital's rule to this function since there is no indeterminateness of 0/0 or oo/oo type
The graph
One‐sided limits
[src]
/ 2\
|-1 - 2*x + 3*x |
lim |---------------|
x->1+| 2 |
\ 1 + x + 4*x /
$$\lim_{x \to 1^+}\left(\frac{3 x^{2} + \left(- 2 x - 1\right)}{4 x + \left(x^{2} + 1\right)}\right)$$
0
$$0$$
= 7.05349503376048e-31
/ 2\
|-1 - 2*x + 3*x |
lim |---------------|
x->1-| 2 |
\ 1 + x + 4*x /
$$\lim_{x \to 1^-}\left(\frac{3 x^{2} + \left(- 2 x - 1\right)}{4 x + \left(x^{2} + 1\right)}\right)$$
0
$$0$$
= -4.21717804841302e-34
= -4.21717804841302e-34
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 1^-}\left(\frac{3 x^{2} + \left(- 2 x - 1\right)}{4 x + \left(x^{2} + 1\right)}\right) = 0$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{3 x^{2} + \left(- 2 x - 1\right)}{4 x + \left(x^{2} + 1\right)}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{3 x^{2} + \left(- 2 x - 1\right)}{4 x + \left(x^{2} + 1\right)}\right) = 3$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{3 x^{2} + \left(- 2 x - 1\right)}{4 x + \left(x^{2} + 1\right)}\right) = -1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{3 x^{2} + \left(- 2 x - 1\right)}{4 x + \left(x^{2} + 1\right)}\right) = -1$$
More at x→0 from the right
$$\lim_{x \to -\infty}\left(\frac{3 x^{2} + \left(- 2 x - 1\right)}{4 x + \left(x^{2} + 1\right)}\right) = 3$$
More at x→-oo
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{3 x^{2} + \left(- 2 x - 1\right)}{4 x + \left(x^{2} + 1\right)}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{3 x^{2} + \left(- 2 x - 1\right)}{4 x + \left(x^{2} + 1\right)}\right) = 3$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{3 x^{2} + \left(- 2 x - 1\right)}{4 x + \left(x^{2} + 1\right)}\right) = -1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{3 x^{2} + \left(- 2 x - 1\right)}{4 x + \left(x^{2} + 1\right)}\right) = -1$$
More at x→0 from the right
$$\lim_{x \to -\infty}\left(\frac{3 x^{2} + \left(- 2 x - 1\right)}{4 x + \left(x^{2} + 1\right)}\right) = 3$$
More at x→-oo
The graph