Questions: Find the limit of the function (if it exists). (If an answer does not exist, enter DNE.) lim (x -> 4) (x^3 - 64) / (x - 4) Write a simpler function that agrees with the given function at all but one point. Use a graphing utility to confirm your result. g(x) = x^2 + 4x + 16

Solution Steps

To find the limit of the function as \( x \) approaches 4, we can simplify the expression by factoring the numerator. The numerator \( x^3 - 64 \) is a difference of cubes, which can be factored using the formula \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \). After factoring, we can cancel out the common factor with the denominator and then substitute \( x = 4 \) to find the limit.

The given function is

\[ \frac{x^3 - 64}{x - 4} \]

The numerator \(x^3 - 64\) is a difference of cubes, which can be factored using the formula \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\). Here, \(a = x\) and \(b = 4\), so we have:

\[ x^3 - 64 = (x - 4)(x^2 + 4x + 16) \]

Substitute the factored form of the numerator back into the original expression:

\[ \frac{(x - 4)(x^2 + 4x + 16)}{x - 4} \]

Cancel the common factor \((x - 4)\) from the numerator and the denominator:

\[ x^2 + 4x + 16 \]

Now, evaluate the limit of the simplified expression as \(x\) approaches 4:

\[ \lim_{x \to 4} (x^2 + 4x + 16) \]

Substitute \(x = 4\) into the simplified expression:

\[ 4^2 + 4 \times 4 + 16 = 16 + 16 + 16 = 48 \]

Final Answer