Q.Show that $\lim_{x \to 4} \dfrac{|x - 4|}{x - 4}$ does not exists.

The key idea is that for a limit to exist at a point, the left-hand limit and the right-hand limit must both exist and be equal. This is crucial for functions involving absolute values, as their definition changes around the point where the expression inside the absolute value becomes zero.

1. Define the function $f(x) = \dfrac{|x - 4|}{x - 4}$.
2. Calculate the right-hand limit (RHL): As $x \to 4^+$, $x > 4$, so $x - 4 > 0$. Thus, $|x - 4| = x - 4$. $\lim_{x \to 4^+} \dfrac{|x - 4|}{x - 4} = \lim_{x \to 4^+} \dfrac{x - 4}{x - 4} = \lim_{x \to 4^+} 1 = 1$.
3. Calculate the left-hand limit (LHL): As $x \to 4^-$, $x < 4$, so $x - 4 < 0$. Thus, $|x - 4| = -(x - 4)$. …

The concept of a limit describes the value a function "approaches" as its input approaches some specific value. For a limit to exist at a particular point, the function must approach the same value whether we approach that point from the left side (values less than the point) or from the right side (values greater than the point). If these two "approaches" lead to different values, then the function does not settle on a single value, and thus the limit does not exist.

This problem involves an absolute value function, $|x-4|$. The definition of an absolute value changes depending on whether the expression inside it is positive or negative. This piecewise nature is crucial for understanding how the function behaves around $x=4$.

In our case, $a = x-4$. So, the function $\frac{|x-4|}{x-4}$ will behave differently depending on whether $x-4$ is positive or negative. This is precisely what we need to examine when approaching $x=4$ from the left and from the right.

Step-by-step solution:

1. Define the function piecewise.

   The expression inside the absolute value is $x-4$.

   • If $x-4 > 0$, which means $x > 4$, then $|x-4| = x-4$.
   • If $x-4 < 0$, which means $x < 4$, then $|x-4| = -(x-4)$. (We do not consider $x-4=0$ because the denominator would be zero, making the function undefined at $x=4$.)

   So, the function $f(x) = \frac{|x-4|}{x-4}$ can be written as:

Simplifying this, we get:

This simplified form clearly shows how the function behaves on either side of $x=4$.

2. Evaluate the left-hand limit (LHL).

The left-hand limit considers values of $x$ approaching $4$ from the left side, meaning $x < 4$.

For $x < 4$, we know that $f(x) = -1$.

Therefore, the left-hand limit is:

Since the limit of a constant is the constant itself:

3. Evaluate the right-hand limit (RHL). The right-hand limit considers values of $x$ approaching $4$ from the right side, meaning $x > 4$. For $x > 4$, we know that $f(x) = 1$. Therefore, the right-hand limit is:

Since the limit of a constant is the constant itself:
$$\lim_{x \to 4^+} f(x) = 1$$ …