The function D defined by $D(x) = egin{cases} 0 & ext{if x is rational} \ 1 & ext{if x is not rational} end{cases}$ is shown in the second video of Week 2 to have the property that $lim_{x o b} D(x)$ does not exist for any real number b. 1. Prove that $lim_{x o infty} D(x)$ also does not exist. 2. Using the function D as a starting point, find a function E such that $lim_{x o b} E(x)$ does not exist for any real number b but $lim_{x o infty} E(x)$ does exist.
Added by Grace H.
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To prove that \( \lim_{x \to \infty} D(x) \) does not exist, we need to show that there is no single value \( L \) such that for every sequence \( \{x_n\} \) with \( x_n \to \infty \), \( D(x_n) \to L \). 1.1. Consider two sequences: - \( \{x_n\} \) where \( x_n Show more…
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