Evaluate: (without using Hopital's rule)
d) lim (6-3x)/(((5x+6)^1/2)-4)
x->2 ^
| square root
the question didn't submit well, here's the problem:
lim (6-3x)/(((5x+6)^1/2)-4)
x->2
Well,
f(2.0001) = -4.8000
f(1.9999) = -4.8000
Looks to me like f(2) = -4.8
To evaluate the limit, we can rationalize the denominator. Rationalizing the denominator means to eliminate any square roots from the denominator by multiplying the expression by a suitable conjugate.
In this case, the denominator is ((5x+6)^(1/2)) - 4. To rationalize it, we need to multiply the numerator and denominator by the conjugate expression: ((5x+6)^(1/2)) + 4.
Multiplying the numerator and denominator by the conjugate expression gives us:
lim(x->2) [(6-3x)*((5x+6)^(1/2)) + 4*(6-3x)] / [((5x+6)^(1/2)) - 4] * [((5x+6)^(1/2)) + 4]
Expanding and simplifying the numerator and denominator, we get:
lim(x->2) [6(5x+6)^(1/2) - 3x(5x+6)^(1/2) + 24 - 12x] / [(5x+6) - 16]
Next, we can simplify further by factoring out (5x+6)^(1/2) from the numerator:
lim(x->2) [(5x+6)^(1/2) * (6 - 3x + 4 - 2x)] / (5x+6 - 16)
Simplifying the expression inside the square brackets, we get:
lim(x->2) [(5x+6)^(1/2) * (-5x + 10)] / (5x - 10)
Now we can cancel out the common factors:
lim(x->2) [(5x+6)^(1/2) * (-1)] / 1
Simplifying the expression further:
lim(x->2) -[(5x+6)^(1/2)]
Finally, substitute x = 2 into the expression:
-[(5(2)+6)^(1/2)] = -[(10+6)^(1/2)]
Therefore, the value of the limit is -[(16)^(1/2)], which is -4.