Lim x->4 (x^2+3x+4)/5x-12
To evaluate the limit as x approaches 4 of the expression (x^2 + 3x + 4) / (5x - 12), we can use direct substitution by plugging in the value x = 4 into the expression. However, this yields an undefined result because it would lead to division by zero.
To overcome this issue, we can try factoring the numerator to see if any common factors cancel out the denominator. Let's factorize the numerator.
The numerator, x^2 + 3x + 4, cannot be factored further, either by finding two binomials that multiply to give this quadratic expression or by applying the quadratic formula. Therefore, factoring doesn't help in simplifying the expression.
Instead, we can try using L'Hopital's rule, which states that if we have an indeterminate form (such as 0/0 or ∞/∞), the limit of the expression can be found by differentiating the numerator and the denominator separately, then taking the limit of the resulting expression.
Let's apply L'Hopital's rule to solve this limit:
Start by finding the derivative of the numerator and the derivative of the denominator.
Numerator derivative:
f'(x) = 2x + 3
Denominator derivative:
g'(x) = 5
Now, let's rewrite the expression with the derivatives:
lim x->4 (2x + 3) / 5
Next, we can perform direct substitution again by plugging in x = 4 into the expression:
(2(4) + 3) / 5 = (8 + 3) / 5 = 11/5
Therefore, the limit of the given expression as x approaches 4 is equal to 11/5.