Answer
See proof below.
Work Step by Step
If $a \gt 0$, note that $|a|=a.$
When x approaches a, being very close to $a$, x is positive, so $|x|=x.$
So, when x approaches a,$ |x|$=$x$ approaches $a$ ,
which is, in this case, equal to $|a|$.
In other words, for $a \gt 0$, $\displaystyle \lim_{x\rightarrow a}|x|=|a|$
If $a \lt 0$, note that $|a|=-a.$
when x approaches a, being very close to $a$, x is negative, so $|x|=-x.$
So, when x approaches a,
$|x|=-x$ (which is now positive) approaches $-a$ (also positive),
which is, in this case, equal to $|a|$.
In other words, for $a \lt 0$, $\displaystyle \lim_{x\rightarrow a}|x|=|a|$
For the case $a=0$, the last exercise showed $\displaystyle \lim_{x\rightarrow 0}|x|=0=|0|.$
Thus, whatever the value of a, it is always that $\displaystyle \lim_{x\rightarrow a}|x|=|a|$
