Find the value of the limit.
lim �ã(x+4) - 2 divided by x
x->0
The 2 is outside of the square root.
The answer is supposed to be 1/4 but I keep getting 0/0 no matter how many times I redo the algebra.
(√(x+4)-2)/x
using l'Hospital's Rule, that gives
(1/(2√(x+4)))/(1) = (1/(2*2))/2 = 1/4
To find the limit, we can first simplify the expression by applying algebraic manipulations. Let's start by simplifying the numerator:
(x + 4)^2 - 4 = x^2 + 8x + 16 - 4 = x^2 + 8x + 12.
Now, let's rewrite the expression with the simplified numerator:
lim (x^2 + 8x + 12) / x
x->0
The expression is still indeterminate at this point, as both the numerator and denominator approach zero as x approaches zero. To further simplify the expression and evaluate the limit accurately, we can factor the numerator:
(x^2 + 8x + 12) = (x + 2)(x + 6).
Now, let's rewrite the expression with the factored numerator:
lim ((x + 2)(x + 6)) / x
x->0
We can see that the x terms in the numerator and denominator cancel out, leaving us with:
lim (x + 6) / 1
x->0
Now, we can directly substitute x = 0 into the expression:
(0 + 6) / 1 = 6 / 1 = 6.
Therefore, the limit of the given expression as x approaches 0 is 6, not 1/4 as mentioned.
If you have been continuously getting an indeterminate form like 0/0, please double-check the algebraic simplifications and calculations to ensure accuracy.