Question 3 (a) Given that \( L=\lim _{x \rightarrow 0} \frac{\sin x}{x} \) exists. Use a table of suitable values to estimate \( L \). (b) Based on your estimation in (a), calculate the limit \[ \lim _{x \rightarrow 0} \frac{x \tan x}{1-\cos x} \]
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To estimate the limit, we can calculate \(\frac{\sin x}{x}\) for values of \(x\) approaching 0 from both the positive and negative sides. \[ \begin{array}{c|c} x & \frac{\sin x}{x} \\ \hline 0.1 & 0.9983 \\ 0.01 & 0.999983 \\ 0.001 & 0.99999983 \\ -0.1 & 0.9983 Show more…
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(a) Complete the table and make a guess about the limit indicated. $$ f(x)=\tan ^{-1}\left(\frac{1}{x}\right) \quad \lim _{x \rightarrow 0^{+}} f(x) $$ $$ \begin{array}{|c|c|c|c|c|c|c|}\hline x & {0.1} & {0.01} & {0.001} & {0.0001} & {0.00001} & {0.000001} \\ \hline f(x) & {} & {} & {} & {} \\ \hline\end{array} $$ (b) Use Figure 1.3 .3 to find the exact value of the limit in part (a).
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