lim x ->-4 (sqrt(x^2+9)-3)/(r+4)
To evaluate the limit, we can plug in the value -4 into the expression and direct substitution method:
lim(x->-4) (sqrt(x^2+9)-3)/(x+4)
= (sqrt((-4)^2+9)-3)/((-4)+4)
= (sqrt(16+9)-3)/0
= (sqrt(25)-3)/0
= (5-3)/0
= 2/0
Since the denominator approaches 0, the limit is undefined.
To find the limit of the given expression as x approaches -4, we can use direct substitution. Let's substitute x = -4 into the expression and simplify:
lim x -> -4 (sqrt(x^2 + 9) - 3)/(x + 4)
= (sqrt((-4)^2 + 9) - 3)/(-4 + 4)
= (sqrt(16 + 9) - 3)/0
As we can see, this expression is indeterminate since we have 0 in the denominator. To further evaluate the limit, we need to rewrite the expression.
Let's rationalize the denominator by multiplying both the numerator and denominator by the conjugate of x + 4, which is √(x^2 + 9) + 3:
lim x -> -4 [(sqrt(x^2 + 9) - 3)/(x + 4)] * [(sqrt(x^2 + 9) + 3)/(sqrt(x^2 + 9) + 3)]
= lim x -> -4 [(sqrt(x^2 + 9)^2 - 3^2)/(x^2 - 4^2)]
= lim x -> -4 (x^2 + 9 - 9)/(x^2 - 16)
= lim x -> -4 x^2/(x^2 - 16)
Now we can cancel out the common factor of x^2:
lim x -> -4 x^2/(x^2 - 16) = lim x -> -4 x/(x - 4)
= (-4^2)/(-4 - 4)
= 16/-8
= -2
Therefore, the limit of the given expression as x approaches -4 is -2.
To evaluate the limit as x approaches -4 of the given expression, we can directly substitute -4 into the expression and simplify.
Let's substitute -4 into the expression:
lim x->-4 (sqrt(x^2 + 9) - 3) / (x + 4)
= (sqrt((-4)^2 + 9) - 3) / (-4 + 4)
= (sqrt(16 + 9) - 3) / (0)
Now, let's simplify the numerator:
sqrt(16 + 9) = sqrt(25) = 5
= (5 - 3) / 0
= 2 / 0
Since division by zero is undefined in mathematics, the limit as x approaches -4 of the given expression is undefined.
Therefore, the answer to lim x -> -4 (sqrt(x^2+9)-3)/(x+4) is undefined.