We are tasked with finding the limit of the difference quotient:

$\lim_{{h \to 0}} \frac{{f(4+h) - f(4)}}{h}$ where $f(x) = |5x - 1|$.

Step 1: Compute $f(4)$

Substitute $x = 4$ into the given function $f(x)$: $f(4) = |5(4) - 1| = |20 - 1| = |19| = 19$

Step 2: Compute $f(4+h)$

We need to evaluate $f(4 + h)$: $f(4+h) = |5(4+h) - 1| = |20 + 5h - 1| = |19 + 5h|$

Step 3: Behavior of $f(4+h)$

Now, we consider two cases for the absolute value $|19 + 5h|$:

• When $h \geq 0$: $f(4+h) = 19 + 5h$
• When $h < 0$: $f(4+h) = -(19 + 5h) = -19 - 5h$

Step 4: Left-Hand and Right-Hand Limits

Right-hand limit $h \to 0^+$:

For $h > 0$, we use $f(4+h) = 19 + 5h$: $\frac{{f(4+h) - f(4)}}{h} = \frac{{(19 + 5h) - 19}}{h} = \frac{{5h}}{h} = 5$

Thus, as $h \to 0^+$, the limit is 5.

Left-hand limit $h \to 0^-$:

For $h < 0$, we use $f(4+h) = -19 - 5h$: $\frac{{f(4+h) - f(4)}}{h} = \frac{{(-19 - 5h) - 19}}{h} = \frac{{-38 - 5h}}{h} = \frac{{-5h}}{h} = -5$

Thus, as $h \to 0^-$, the limit is $-5$.

Step 5: Conclusion

The left-hand and right-hand limits are not equal. Therefore, the limit does not exist. The limit of the difference quotient is undefined.

Would you like more details or have any questions?

Here are 5 related questions for further understanding:

1. What is the geometric interpretation of a derivative when a limit doesn't exist?
2. Can a function with an absolute value have a smooth derivative everywhere?
3. How do piecewise functions affect the calculation of limits?
4. What role does the sign of $h$ play in limits involving absolute value functions?
5. What conditions must be met for the derivative to exist at a point of non-differentiability?

Tip: When dealing with absolute value functions, always break the function into cases to understand its behavior near critical points like zero.