Evaluate the following limits:
(I) Lim. (x+1)/[1-sqrt(4+3x)]
X >-1
(II) Lim. (2-3x-5x^2)/(1+2x^2)
X >Infinite
To evaluate the given limits, let's go step by step:
(I) Limit: (x+1)/(1-sqrt(4+3x)), as x approaches -1.
To find the limit, we need to substitute the value that x approaches into the expression and see if it converges to a finite value. In this case, x approaches -1, so we can substitute -1 into the expression:
(x+1)/(1-sqrt(4+3x)) = (-1+1)/(1-sqrt(4+3*(-1)))
Simplifying further:
(0)/(1-sqrt(4-3)) = 0/(-2) = 0
Therefore, the limit is 0.
(II) Limit: (2-3x-5x^2)/(1+2x^2), as x approaches infinity.
To find the limit as x approaches infinity, we examine the behavior of the expression as x gets larger and larger. We can do this by considering the highest power of x in the numerator and denominator.
In this case, the highest powers are x^2 in the numerator and denominator. Dividing the entire expression by x^2, we have:
(2/x^2 - 3/x - 5)/(1/x^2 + 2)
As x approaches infinity, 2/x^2, 3/x, and 1/x^2 all approach 0. Hence, the expression simplifies to:
(0 - 0 - 5)/(0 + 2) = -5/2
Therefore, the limit is -5/2.