Single Variable Calculus

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Single Variable Calculus, Chapter 2, 2.2, Section 2.2, Problem 37

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(a) Evaluate the function $f(x) = x^2 - (2^x/1000)$ for $x = 1 $, 0.8, 0.6, 0.4, 0.2, 0.1 and 0.05. and guess the value of $\displaystyle \lim \limits_{x \to 0} \left( x^2 - \frac{2^x}{1000} \right)$

$ \begin{array}{|c|c|} \hline x & f(x)\\ \hline 1 & 0.998\\ 0.8 & 0.638\\ 0.6 & 0.358\\ 0.4 & 0.159\\ 0.2 & 0.039\\ 0.1 & 0.009\\ 0.05 & 0.001\\ \hline \end{array} $

Based from the vlaues we obtain in the table; as $x$ approaches 0, the value of $f(x)$ approaches 0 as well. Therefore $\displaystyle \lim \limits_{x \to 0} \left( x^2 - \frac{2^x}{1000} \right) = 0$

(b) Evaluate $f(x)$ for $x = $ 0.04, 0.02, 0.01, 0.005, 0.003 and 0.001. Guess again

$ \begin{array}{|c|c|} \hline x & f(x)\\ \hline 0.04 & 0.00057\\ 0.02 & -0.00061\\ 0.01 & -0.00091\\ 0.005 & -0.00098\\ 0.003 & -0.00099\\ 0.001 & -0.001\\ \hline \end{array} $

Based from the values we obtain from the table; The $\lim\limits_{x \to 0} f(x)$ seems to have a value of -0.001.

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