limit as x approaches negative infinity for the function (-x)/(sqr root (x^2-36))
sqrt (x^2-36) -> x as x-> -infinity
The limit is therefore -x/|x| -> 1
when x-> -infinity
Check: When x = -1000,
(-x)/(sqr root (x^2-36)) = 1000/999.9 = 1.00002
When x = 100, (-x)/(sqr root (x^2-36)) = 1000/998.2 = 1.0018
To find the limit as x approaches negative infinity for the function (-x)/(√(x^2-36)), we need to determine the behavior of the function as x gets infinitely small (negative).
First, let's simplify the expression (-x)/(√(x^2-36)):
As x approaches negative infinity, √(x^2-36) approaches √(x^2) = |x|, since the square root of a positive number is always positive. Therefore, we can simplify the expression as (-x)/|x|.
Now, we can divide both numerator and denominator by |x| (assuming x ≠ 0 since we can't divide by zero):
(-x)/|x| = -1
Thus, as x approaches negative infinity, the given function approaches -1.
To find the limit as x approaches negative infinity for the function (-x)/(sqrt(x^2-36)), we can use some algebraic manipulations.
First, let's simplify the expression inside the square root: x^2 - 36.
As x approaches negative infinity, x^2 becomes very large, and the negative 36 becomes negligible compared to x^2. So we can approximate x^2 - 36 as x^2.
Now, the function becomes (-x)/(sqrt(x^2)). Notice that the square root of x^2 is simply the absolute value of x, |x|.
Since we are approaching negative infinity, x is negative, so |x| is equal to -x.
Therefore, the function becomes (-x)/(-x), and the negative signs cancel out, leaving us with 1.
So, the limit as x approaches negative infinity for the function (-x)/(sqrt(x^2-36)) is equal to 1.