Lim (x^4-2x+10)/(x^3-2x+1)x>1

Предыдущий
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вопрос

To solve this inequality, first simplify the expression inside the parentheses:

(x^4-2x+10)/(x^3-2x+1) = (x^4-2x+10)/(x-1)^2

Next, we have the inequality:
(x^4-2x+10)/(x-1)^2 > 1

Now, we can multiply both sides by (x-1)^2 to get rid of the denominator:
x^4 - 2x + 10 > (x-1)^2

Expand the right side:
x^4 - 2x + 10 > x^2 - 2x + 1

Subtract x^2 - 2x + 1 from both sides to set the inequality to zero:
x^4 - x^2 + 9 > 0

Factor the left side:
(x^2 - 3)(x^2 + 3) > 0

Now find the critical points by setting each factor to zero:
x^2 - 3 = 0
x^2 = 3
x = ±√3

Since this is a quadratic inequality, we can use the critical points to determine the intervals where the inequality is true.
Testing a point in each interval:
For x < -√3, choose x = -4:
((-4)^2 - 3)((-4)^2 + 3) > 0
(16 - 3)(16 + 3) > 0
13 * 19 > 0
247 > 0
True for x < -√3

For -√3 < x < √3, choose x = 0:
((0)^2 - 3)((0)^2 + 3) > 0
(-3)(3) > 0
-9 > 0
False for -√3 < x < √3

For x > √3, choose x = 4:
((4)^2 - 3)((4)^2 + 3) > 0
(16 - 3)(16 + 3) > 0
13 * 19 > 0
247 > 0
True for x > √3

Therefore, the solution to the inequality is x < -√3 or x > √3.