Let's solve the limit step by step:
Given expression:
[tex]\[ \lim_{x \rightarrow \infty} \sqrt{x}(\sqrt{x} - \sqrt{x} - a) \][/tex]
First, simplify the expression inside the parenthesis:
[tex]\[ \sqrt{x} - \sqrt{x} - a = 0 - a = -a \][/tex]
Now, substitute this back into the original expression:
[tex]\[ \lim_{x \rightarrow \infty} \sqrt{x}(-a) \][/tex]
This simplifies to:
[tex]\[ \lim_{x \rightarrow \infty} -a \sqrt{x} \][/tex]
Next, analyze the behavior of [tex]\(\sqrt{x}\)[/tex] as [tex]\(x\)[/tex] approaches infinity. The square root of [tex]\(x\)[/tex], or [tex]\(\sqrt{x}\)[/tex], also approaches infinity as [tex]\(x\)[/tex] becomes infinitely large.
Therefore:
[tex]\[ \lim_{x \rightarrow \infty} -a \sqrt{x} \][/tex]
Given that [tex]\(a\)[/tex] is a non-zero constant, [tex]\( -a \sqrt{x} \)[/tex] will grow without bound in the negative direction because it is being multiplied by [tex]\(-a\)[/tex], which introduces a negative sign.
As a result, the expression:
[tex]\[ \lim_{x \rightarrow \infty} -a \sqrt{x} \][/tex]
approaches [tex]\(-\infty\)[/tex].
Therefore:
[tex]\[ \lim_{x \rightarrow \infty} \sqrt{x}(\sqrt{x} - \sqrt{x} - a) = -\infty \][/tex]
So, the final answer is:
[tex]\[ -\infty \][/tex]
