lim x→−12 sqrt(x^2 +25 −13)/x+12
I will assume you meant the numerator to be
( √(x^2+25) - 13 ) , so
lim ( √(x^2+25) - 13 )/(x+12) , as x-->-12
= lim ( √(x^2+25) - 13 )/(x+12) * ( √(x^2+25) + 13 )/( √(x^2+25) + 13 )
= lim( x^2 + 25 - 169)/( ( √(x^2+25) + 13 )(x-12) )
= lim (x^2 - 144)/( ( √(x^2+25) + 13 )(x-12) )
= lim (x+12)(x-12)/( ( √(x^2+25) + 13 )(x-12) )
= lim (x-12)/( √(x^2+25) + 13 )
= -24/√26
= -12/13
express 9,234, in place value notation:
To find the limit of the given expression as x approaches -12, we can simplify the expression and then substitute the value of -12.
Let's start by simplifying the expression:
lim x→−12 sqrt(x^2 +25 −13)/(x+12)
Since the expression inside the square root is a perfect square, we can simplify it further:
lim x→−12 sqrt((x^2 - 4x + 4) + 21)/(x+12)
Now, we can rewrite the expression inside the square root as a square of a binomial:
lim x→−12 sqrt((x-2)^2 + 21)/(x+12)
Taking the square root of the expression:
lim x→−12 ((x-2)^2 + 21)^(1/2)/(x+12)
As x approaches -12, both the numerator and the denominator approach zero. This indicates that we have an indeterminate form and we need to apply a different approach to evaluate the limit.
To simplify the expression further, we can multiply both the numerator and denominator by the conjugate of the denominator, (x-12):
lim x→−12 ((x-2)^2 + 21)^(1/2)/(x+12) * ((x-12)/(x-12))
Simplifying:
lim x→−12 (x-2)^2 + 21)/(x+12)*(x-12)
Expanding the numerator:
lim x→−12 (x^2 - 4x + 4 + 21)/(x+12)*(x-12)
Simplifying:
lim x→−12 (x^2 - 4x + 25)/(x+12)
Now, we can substitute x = -12 into the expression:
((-12)^2 - 4*(-12) + 25)/(-12+12)
= (144 + 48 + 25)/(0)
= 217/0
Since the denominator is zero, the limit does not exist.
To evaluate the limit,
lim x→-12 sqrt(x^2 + 25 -13)/(x + 12),
we can simplify the expression inside the square root first. Notice that x^2 + 25 - 13 can be rewritten as x^2 + 12^2 - 13, which is a perfect square trinomial. We can simplify it using the form "a^2 - b^2 = (a + b)(a - b)".
x^2 + 12^2 - 13 = (x^2 + 12^2) - 13 = (x + 12)(x - 12).
Now let's rewrite the limit expression with the simplified form:
lim x→-12 sqrt((x + 12)(x - 12))/(x + 12).
We can observe that (x + 12) appears both in the denominator and inside the square root. Since the numerator and denominator share the common factor (x + 12), we can cancel it out:
lim x→-12 sqrt(x - 12).
Now we just need to evaluate this new limit expression. As x approaches -12, x - 12 approaches -12 - 12 = -24.
Therefore, the simplified limit expression becomes:
lim x→-12 sqrt(x - 12) = sqrt(-24).
However, the square root of a negative number is not a real number, so this limit is undefined.