limx→4 4x^2−15x−4/x−4.
using l'Hospital's Rule, it's the same limit as
(8x-15)/1 = 17
To find the limit of a function as x approaches a certain value, you can try direct substitution. However, in this case, if you directly substitute x = 4 into the given function, you would get division by zero, which is undefined.
To evaluate the limit in this case, you can simplify the function using factoring or polynomial division.
Step 1: Factor the numerator
Rewrite the numerator of the function: 4x^2 - 15x - 4
To factor this quadratic expression, we need to find two numbers whose product is equal to the constant term (-4) and whose sum is equal to the coefficient of the linear term (-15).
After some analysis, we can find that the numbers are -16 and 1:
4x^2 - 15x - 4 = (4x + 1)(x - 4)
Step 2: Cancel out the common factor
Now that we have factored the numerator, we can cancel out the common factor (x - 4) with the denominator (x - 4):
f(x) = (4x + 1)
Step 3: Evaluate the limit
Now that the function has been simplified, we can directly substitute x = 4 into the simplified function:
limx→4 (4x^2 - 15x - 4)/(x - 4) = limx→4 (4x + 1) = 4(4) + 1 = 17
Therefore, the limit of the function as x approaches 4 is 17.