lim x->16 4- sqrt(x) / x-16
Did you realize that
x-16 = (√x+4)(√x-4) ?
then your limit question simplifies to
lim (4-√x)/( (√x+4)(√x-4) ) as x --> 16
= lim -1/(√x+4) as x --> 16
= -1/8
To find the limit of the expression (4 - sqrt(x)) / (x - 16) as x approaches 16, we can simplify it by rationalizing the numerator.
Step 1: Rationalize the numerator:
Multiply the numerator and denominator by the conjugate of the numerator, which is (4 + sqrt(x)). This allows us to eliminate the square root in the numerator.
[(4 - sqrt(x)) / (x - 16)] * [(4 + sqrt(x)) / (4 + sqrt(x))]
Step 2: Simplify the numerator:
Using the difference of squares, we get:
[(4^2 - (sqrt(x))^2) / (x - 16)] * [(4 + sqrt(x)) / (4 + sqrt(x))]
Simplifying:
[(16 - x) / (x - 16)] * [(4 + sqrt(x)) / (4 + sqrt(x))]
Step 3: Cancel out the common factors:
The numerator (16 - x) and the denominator (x - 16) cancel each other out:
[(4 + sqrt(x)) / 1] = (4 + sqrt(x))
Step 4: Evaluate the limit:
Now, we can find the limit as x approaches 16:
lim x→16 (4 - sqrt(x)) / (x - 16) = lim x→16 (4 + sqrt(x)) = 4 + sqrt(16) = 4 + 4 = 8
Therefore, the limit of the expression (4 - sqrt(x)) / (x - 16) as x approaches 16 is 8.
To find the limit of the given expression as x approaches 16, you can use algebraic manipulation and apply limit properties. Here's how:
Step 1: Rewrite the expression:
lim x->16 of (4 - sqrt(x))/(x - 16)
Step 2: Rationalize the numerator:
Multiply the numerator and denominator by the conjugate of the numerator, which is (4 + sqrt(x)). This helps to eliminate the square root in the numerator.
lim x->16 of [(4 - sqrt(x))(4 + sqrt(x))] / [(x - 16)(4 + sqrt(x))]
Step 3: Simplify the numerator:
Using the difference of squares formula, (a-b)(a+b) = a^2 - b^2, we have:
lim x->16 of (16 - x) / [(x - 16)(4 + sqrt(x))]
Step 4: Simplify further:
We can cancel out the (x - 16) term in the numerator and denominator:
lim x->16 of -(16 - x) / (4 + sqrt(x))
Step 5: Evaluate the limit:
Plugging 16 into the expression:
lim x->16 of -(16 - 16) / (4 + sqrt(16))
lim x->16 of 0 / (4 + 4)
lim x->16 of 0 / 8
= 0
Therefore, the limit of the given expression as x approaches 16 is 0.