Evaluate the limit, if it exists.
lim x->-4 (x+4)/(x^3 + 64)
To evaluate the limit as x approaches -4, we substitute -4 into the expression:
lim x->-4 (x+4)/(x^3 + 64) = (-4 + 4)/((-4)^3 + 64) = 0/(64 - 64) = 0/0
Since we get 0/0, this is an indeterminate form. To further evaluate the limit, we can factor the denominator:
lim x->-4 (x+4)/(x^3 + 64) = (x+4)/((x+4)(x^2 - 4x + 16))
We can cancel out the common factor of (x+4):
lim x->-4 1/(x^2 - 4x + 16)
Now we can substitute -4 into the expression:
lim x->-4 1/((-4)^2 - 4(-4) + 16) = 1/(16 + 16 + 16) = 1/48
Therefore, the limit of the expression as x approaches -4 is 1/48.
To evaluate the limit of (x + 4)/(x^3 + 64) as x approaches -4, we substitute x = -4 into the expression and simplify:
lim x->-4 (x + 4) / (x^3 + 64)
= (-4 + 4) / ((-4)^3 + 64)
= 0 / (64 - 64)
= 0 / 0
When we encounter the 0 / 0 form, it indicates an indeterminate form, and further evaluation is needed. To determine the limit, we can factorize the denominator using the difference of cubes formula:
x^3 + 64 = (x + 4)(x^2 - 4x + 16)
After factoring, the denominator cancels out with the numerator, leaving us with:
lim x->-4 1 / (x^2 - 4x + 16)
Now, we can substitute x = -4 into the expression and simplify further:
lim x->-4 1 / ((-4)^2 - 4(-4) + 16)
= 1 / (16 + 16 + 16)
= 1 / 48
Therefore, the limit of (x + 4) / (x^3 + 64) as x approaches -4 is 1 / 48.
To evaluate the limit, we substitute the value that x is approaching into the function and simplify to see if we get a finite value. In this case, we are evaluating the limit as x approaches -4.
Substituting -4 into the function, we get:
lim x->-4 (x+4)/(x^3 + 64) = (-4 + 4)/((-4)^3 + 64) = 0/0
We obtained an indeterminate form (0/0) in this case. To evaluate this type of limit, we can use algebraic manipulation to simplify the expression further.
Factoring the denominator using the difference of cubes formula, we have:
(-4)^3 + 64 = (-4 + 4)(16 - 16 + 64) = 0 * 64 = 0
Now, let's simplify the expression by canceling out the common factors:
lim x->-4 (x+4)/(x^3 + 64) = (x+4)/[(x+4)(16-4x+x^2)]
Now we can cancel out the common factor (x+4):
lim x->-4 (x+4)/(x^3 + 64) = 1/(16-4x+x^2)
Since the denominator does not have any common factors with the numerator and the denominator is nonzero at x = -4, we can substitute x = -4 into the simplified expression to find the limit:
lim x->-4 1/(16-4x+x^2) = 1/(16+16+16) = 1/48
Therefore, the limit as x approaches -4 of (x+4)/(x^3 + 64) is 1/48, if it exists.