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Let's break down the two limits: ### (i) \(\lim_{x\to\infty} \left( \frac{x-3}{x-1} \right)^x\) We want to simplify this expression and evaluate the limit. First, let's rewrite the fraction inside the parentheses: \[ \frac{x-3}{x-1} = 1 - \frac{2}{x-1} \] So the expression becomes: \[ \left( 1 - \frac{2}{x-1} \right)^x \] This resembles the standard limit form for \( (1 + \frac{a}{n})^n \) as \( n \to \infty \), which approaches \( e^a \). We ________ ________ _______ ______ ____ ______ ______ _________ __________ ______.
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