find lim x->0 3- sqrt x+9/-x
In general, when there is a sum or difference of square-root terms, multiplying both numerator and denominator by the conjugate is a good strategy.
Lim (3- sqrt( x+9))/(-x )
x->0
= Lim (3-sqrt(x+9))(3+sqrt(x+9))/[(-x)(3+sqrt(x+9)]
x->0
= Lim (9-(x+9))/[(-x)(3+sqrt(x+9)]
x->0
= Lim (-x)/[(-x)(3+sqrt(x+9)]
x->0
= Lim 1/(3+sqrt(x+9)
x->0
=1/6
To find the limit as x approaches 0 of (3 - √(x + 9)) / (-x), we can use algebraic manipulation and the concept of limits.
First, let's simplify the expression by rationalizing the numerator.
Multiply both the numerator and denominator by the conjugate of the numerator, which is (3 + √(x + 9)), to eliminate the square root:
((3 - √(x + 9)) / (-x)) * ((3 + √(x + 9)) / (3 + √(x + 9)))
This simplifies to:
(3^2 - (√(x + 9))^2) / (-x(3 + √(x + 9)))
Simplifying further:
(9 - (x + 9)) / (-x(3 + √(x + 9)))
(-x) / (-x(3 + √(x + 9)))
Now, we can cancel out the common factors of -x:
1 / (3 + √(x + 9))
Now, we can substitute 0 into the expression:
1 / (3 + √(0 + 9))
1 / (3 + 3)
1 / 6
Therefore, the limit as x approaches 0 of (3 - √(x + 9)) / (-x) is equal to 1/6.