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(B) We need to evaluate the limit $L = \lim _{x \rightarrow 0} x^7 \left[ \frac{1}{x^3} \right]$.Using the property of the greatest integer function,$

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$0$

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(B) We need to evaluate the limit $L = \lim _{x \rightarrow 0} x^7 \left[ \frac{1}{x^3} \right]$.Using the property of the greatest integer function,$\frac{1}{x^3} - 1 Case $1$: $x > 0$. Multiplying by $x^7$ (which is positive),we get $x^7 \left( \frac{1}{x^3} - 1 \right) $x^4 - x^7 As $x \rightarrow 0^+$,both $x^4 - x^7 \rightarrow 0$ and $x^4 \rightarrow 0$. By the Squeeze Theorem,the limit is $0$.Case $2$: $x $x^4 \le x^7 \left[ \frac{1}{x^3} \right] As $x \rightarrow 0^-$,both $x^4 \rightarrow 0$ and $x^4 - x^7 \rightarrow 0$. By the Squeeze Theorem,the limit is $0$.Since the left-hand limit and right-hand limit are both $0$,the limit is $0$.

If $[.]$ denotes the greatest integer function,then $\lim _{x \rightarrow 0} x^7 \left[ \frac{1}{x^3} \right]$ is equal to

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