$\"\"$

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Let $\"\"$ and $\"\"$ be a real numbers and let $\"\"$ be a positive integer.

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$\"\"$ .

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Consider $\"\"$ .

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Limit of $\"\"$ as $\"\"$ approaches $\"\"$ does not depend on the value of $\"\"$ at $\"\"$ .

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The function $\"\"$ .

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$\"\"$

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Now we prove the limit $\"\"$ using $\"\"$ definition.

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To show that for each $\"\"$ , there exists a $\"\"$ such that $\"\"$ , whenever $\"\"$ .

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Consider $\"\"$

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$\"\"$

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Consider $\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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Observe the relationship between two absolute values $\"\"$ and $\"\"$ .

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$\"\"$ inequality is always true irrespective of values of $\"\"$ .

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Therefore $\"\"$ then $\"\"$ for any values of $\"\"$ .

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Hence $\"\"$ for all values of $\"\"$ .

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$\"\"$

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$\"\"$ .