A) \[e\]
B) \[x\]
C) \[y\]
D) None of these
Correct Answer: C
Solution :
We have \[\underset{n\to \infty }{\mathop{\lim }}\,\,{{({{x}^{n}}+{{y}^{n}})}^{1/n}}=y\,\,\underset{n\to \infty }{\mathop{\lim }}\,\,{{\left( 1+{{\left( \frac{x}{y} \right)}^{n}} \right)}^{1/n}}\] \[=y\underset{n\to \infty }{\mathop{\lim }}\,{{\left[ 1+{{\left( \frac{x}{y} \right)}^{n}} \right]}^{{{\left( \frac{y}{x} \right)}^{n}}.\frac{1}{n}.{{\left( \frac{x}{y} \right)}^{n}}}}\] \[=y\underset{n\to \infty }{\mathop{\lim }}\,\,\,{{\left[ {{\left( 1+{{\left( \frac{x}{y} \right)}^{n}} \right)}^{{{\left( \frac{y}{x} \right)}^{n}}.}} \right]}^{\frac{1}{n}.{{\left( \frac{x}{y} \right)}^{n}}}}\] \[=y{{e}^{0}}=y\], \[\left[ \because \,\,\frac{x}{y}<1\,\Rightarrow \,\,{{\left( \frac{x}{y} \right)}^{n}}\to 0\,\,\text{as}\,\,n\to \infty \right]\].
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