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  • Using the product law for limits evaluate <img alt="\lim_{y\rightarrow m}\left ( \frac{y+a}{y-b} \right )\times \lim_{y\rightarrow m}\left ( \frac{y-a}{y+b} \right )\times \lim_{y\rightarrow m

Using the product law for limits evaluate \lim_{y\rightarrow m}\left ( \frac{y+a}{y-b} \right )\times \lim_{y\rightarrow m}\left ( \frac{y-a}{y+b} \right )\times \lim_{y\rightarrow m}\left ( \frac{y^{2}+a^{2}}{y^{2}+b^{2}} \right )

Option: 1

\frac{m^{4}-a^{4}}{m^{4}-b^{4}}


Option: 2

\frac{m^{2}-a^{4}}{m^{4}-b^{4}}


Option: 3

\frac{m^{4}-a^{4}}{m^{2}-b^{4}}


Option: 4

\frac{m^{4}-b^{4}}{m^{4}-a^{4}}


Answers (1)

best_answer

Note that the “Product law for limits” states that \lim_{x\rightarrow a}f\left ( x \right )\times\lim_{x\rightarrow a}g\left ( x \right )= \lim_{x\rightarrow a}\left [ f\left ( x \right )\times g\left ( x \right ) \right ].

Now, applythe “Sum law for limits” to evaluate the provided limit in the following way.

\lim _{y \rightarrow m}\left(\frac{y+a}{y-b}\right) \times \lim _{y \rightarrow m}\left(\frac{y-a}{y+b}\right) \times \lim _{y \rightarrow m}\left(\frac{y^2+a^2}{y^2+b^2}\right)

=\lim _{y \rightarrow m}\left[\left(\frac{y+a}{y-b}\right) \times\left(\frac{y-a}{y+b}\right) \times\left(\frac{y^2+a^2}{y^2+b^2}\right)\right]
=\lim _{y \rightarrow m}\left[\left\{\left(\frac{y+a}{y-b}\right) \times\left(\frac{y-a}{y+b}\right)\right\} \times\left(\frac{y^2+a^2}{y^2+b^2}\right)\right]
=\lim _{y \rightarrow m}\left[\left(\frac{y^2-a^2}{y^2-b^2}\right) \times\left(\frac{y^2+a^2}{y^2+b^2}\right)\right]
=\lim _{y \rightarrow m}\left[\left(\frac{y^4-a^4}{y^4-b^4}\right)\right]
=\frac{m^4-a^4}{m^4-b^4}
 

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Nehul

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