can someone please explain why the answer to this is negative infinity? I keep getting positive.
lim
x--> - infinity x^3-2/x^2+x
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Just get a feel for the numbers
x ---> - infinity
for very large negative numbers , x^3 - 2 becomes "
"hugely negative"
but x^2 + x becomes + "very large"
since -/+ = - , and the numerator is larger than the denominator by a factor of x,
the answer is -negative infinity
or
(x^3 - 2)/(x^2 + x) = x - 1 + (x+2)/(x^2+x)
lim (x^3-2)/(x^2+x) as x---> -∞
= lim (x-1) + lim (x+2)/(x^2+x) as x---> -∞
since intuitively we can see that lim (x+2)/(x^2+x) --> 0 as x ---> -∞
we are left with lim x-1 as x -->-∞
which is -∞
To determine the limit as x approaches negative infinity for the expression (x^3-2)/(x^2+x), we need to evaluate the expression as x gets extremely small and negative.
Let's first simplify the expression by factoring out the highest power of x in the numerator and denominator:
(x^3-2)/(x^2+x) = (x^2(x-2))/(x(x+1))
Now we can cancel out the common factor of x:
(x-2)/(x+1)
As x approaches negative infinity, the numerator (x-2) becomes more and more negative, while the denominator (x+1) becomes more and more negative.
Since both the numerator and denominator are heading towards negative infinity and the denominator is growing faster, the expression (x-2)/(x+1) approaches -infinity.
Hence, the limit as x approaches negative infinity for the given expression is -∞.