Question

if (a
eq0), then (lim_{x
ightarrow a}\frac{x^{2}-a^{2}}{x^{4}-a^{4}}) is

Explanation:

Step1: Factorize the expressions

We know that \(x^{4}-a^{4}=(x^{2} - a^{2})(x^{2}+a^{2})\). So, \(\lim_{x
ightarrow a}\frac{x^{2}-a^{2}}{x^{4}-a^{4}}=\lim_{x
ightarrow a}\frac{x^{2}-a^{2}}{(x^{2}-a^{2})(x^{2}+a^{2})}\).

Step2: Cancel out the common factor

Since \(x
eq a\) (in the limit - sense), we can cancel out the non - zero factor \(x^{2}-a^{2}\) (because \(a
eq0\)). Then we get \(\lim_{x
ightarrow a}\frac{1}{x^{2}+a^{2}}\).

Step3: Substitute \(x = a\)

Substitute \(x=a\) into \(\frac{1}{x^{2}+a^{2}}\), we have \(\frac{1}{a^{2}+a^{2}}=\frac{1}{2a^{2}}\).

Answer:

\(\frac{1}{2a^{2}}\)