Question

find \\(\lim\limits_{h \to 0} \frac{f(x + h) - f(x)}{h}\\) for the given function and value of \\(x\\).
\\(f(x) = -5x - 3\\), \\(x = 5\\)

the \\(\lim\limits_{h \to 0} \frac{f(x + h) - f(x)}{h}\\) for \\(f(x) = -5x - 3\\), \\(x = 5\\) is \\(\square\\).
(type an integer or a simplified fraction.)

Explanation:

Step1: Compute \( f(x + h) \)

Given \( f(x) = -5x - 3 \), substitute \( x + h \) into the function:
\( f(x + h) = -5(x + h) - 3 = -5x - 5h - 3 \)

Step2: Compute \( f(x + h) - f(x) \)

Substitute \( f(x + h) \) and \( f(x) \) into the expression:
\( f(x + h) - f(x) = (-5x - 5h - 3) - (-5x - 3) \)
Simplify the expression:
\( = -5x - 5h - 3 + 5x + 3 = -5h \)

Step3: Compute the limit

Now, find \( \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \) by substituting \( f(x + h) - f(x) = -5h \):
\( \lim_{h \to 0} \frac{-5h}{h} \)
Cancel out \( h \) (for \( h
eq 0 \)):
\( \lim_{h \to 0} -5 \)
As \( h \) approaches 0, the limit is -5. The value of \( x = 5 \) does not affect the result here because the derivative of a linear function \( f(x) = mx + b \) is constant \( m \), regardless of \( x \).

Answer:

-5