Evaluate the following limit:
lim x --> negative infinity
9 square root over x^2-1
divided by
x + 4
I assume you want
Lim 9 sqrt(x^2-1)/(x+4)
x-> -inf
When x becomes very large and negative, the numerator approaches 9|x| (assuming you take only the positive square root) and the denominator approaches x. The fraction therefore approaches -9.
To evaluate the given limit, we can use the concept of the highest-powered terms when dealing with limits at infinity. Here's the step-by-step solution:
Step 1: Identify the highest-powered terms in the numerator and denominator:
In the numerator, we have a square root of (x^2 - 1), while in the denominator, there is only x.
Step 2: Divide both the numerator and denominator by the highest power of x:
Dividing the entire expression by x, we get:
lim x --> negative infinity (9√((x^2 - 1)/(x^2))/(x/x + 4/x))
Simplifying further:
lim x --> negative infinity (9√(1 - 1/x^2)/(1 + 4/x))
Step 3: Evaluate the limit as x approaches negative infinity:
As x approaches negative infinity, 1/x approaches 0. Thus, we can make the following substitutions:
lim x --> negative infinity (9√(1 - 1/x^2)/(1 + 4/x))
lim x --> negative infinity (9√(1 - 0^2)/(1 + 0))
lim x --> negative infinity (9√(1)/(1))
Step 4: Simplify to get the final answer:
Since the square root of 1 is 1, we have:
lim x --> negative infinity (9(1)/(1))
lim x --> negative infinity (9/1)
The final answer is 9.