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 limits, let's go step by step:
(I) Lim. (x+1) / [1 - sqrt(4+3x)], x > -1:
To find the limit as x approaches a certain value, in this case, x > -1, we need to determine how the function behaves as x gets closer and closer to that value.
1. Check if the denominator becomes zero: In this case, the denominator 1 - sqrt(4+3x) cannot become zero since the square root of a positive number is always positive.
2. Simplify the expression: To simplify the expression, multiply both the numerator and denominator by the conjugate of the denominator, which is 1 + sqrt(4+3x). This will help us eliminate the square root from the denominator.
(x+1) * (1 + sqrt(4+3x)) / [1 - sqrt(4+3x)] * (1 + sqrt(4+3x))
= (x+1)(1 + sqrt(4+3x))/(1 - (4+3x))
= (x+1)(1 + sqrt(4+3x))/(-3x - 3)
3. Now, let's simplify further and cancel out any common factors:
(x+1)(1 + sqrt(4+3x))/(-3x - 3)
= -(x+1)(1 + sqrt(4+3x))/(3x + 3)
4. Evaluate the limit: By substituting x > -1, we can evaluate the limit by directly substituting x with the value:
Lim. (x+1)(1 + sqrt(4+3x))/(3x + 3) as x approaches -1 from the right side
Substituting x = -1:
(-1+1)(1 + sqrt(4+3(-1)))/(3(-1) + 3)
= 0/0 (indeterminate form)
When we get an indeterminate form (0/0 or ∞/∞), we can apply L'Hospital's Rule, which states that if the limit of the quotient of two functions is an indeterminate form, then the limit of their derivatives is the same. Let's differentiate the numerator and denominator separately:
Numerator: d/dx [(x+1)(1 + sqrt(4+3x))] = 1 + sqrt(4+3x) + (x+1) * [(3/2)(1+3x)^(-1/2)*(3)]
Denominator: d/dx [(3x + 3)] = 3
Now, let's compute the limit of the derivatives:
Lim (1 + sqrt(4+3x) + (x+1) * [(3/2)(1+3x)^(-1/2)*(3)]) / 3 as x approaches -1
= (1 + sqrt(4+3(-1)) + (-1+1) * [(3/2)(1+3(-1))^(-1/2)*(3)]) / 3
= (1 + sqrt(1) + (0) * [(3/2)(1-3)^(1/2)*(3)]) / 3
= (1 + 1 + 0) / 3
= 2/3
Therefore, Lim (x+1) / [1 - sqrt(4+3x)] as x approaches -1 from the right side is equal to 2/3.
(II) Lim (2-3x-5x^2)/(1+2x^2) as x approaches Infinite:
To evaluate this limit, we need to determine the behavior of the function as x becomes extremely large.
1. Compare powers of x: The highest power of x in the numerator is x^2, and the highest power of x in the denominator is x^2. In this case, we can compare the coefficients of the highest power of x to determine the limit.
Lim (2-3x-5x^2)/(1+2x^2) as x approaches Infinite
Since the denominator has the same degree as the numerator, divide both the numerator and denominator by x^2:
= Lim (2/x^2 - 3/x - 5)/(1/x^2 + 2)
2. Evaluate the limit:
As x approaches Infinite, the terms with 1/x^2 and 1/x tend to zero, and we are left with:
= Lim (0 - 0 - 5)/(0 + 2)
= Lim -5/2
= -5/2
Therefore, Lim (2-3x-5x^2)/(1+2x^2) as x approaches Infinite is equal to -5/2.