Chapter 2: Q8E (page 77)
Differentiate
\(g\left( x \right) = \left( {x + {\bf{2}}\sqrt x } \right){e^x}\)
The derivative of g is \({e^x}\left( {x + 2\sqrt x + \frac{1}{{\sqrt x }} + 1} \right)\).
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Use equation 5 to find \(f'\left( a \right)\) at the given number \(a\).
\(f\left( x \right) = \frac{{\bf{1}}}{{\sqrt {{\bf{2}}x + {\bf{2}}} }}\), \(a = {\bf{1}}\)
To prove that sine is continuous, we need to show that \(\mathop {\lim }\limits_{x \to a} \sin x = \sin a\) for every number a. By Exercise 65 an equivalent statement is that
\(\mathop {\lim }\limits_{h \to 0} \sin \left( {a + h} \right) = \sin a\)
Use (6) to show that this is true.
19-32 Prove the statement using the \(\varepsilon \), \(\delta \)definition of a limit.
24. \(\mathop {{\bf{lim}}}\limits_{x \to a} c = c\)
Find \(f'\left( a \right)\).
\(f\left( t \right) = \frac{{\bf{1}}}{{{t^{\bf{2}}} + {\bf{1}}}}\)
19-32: Prove the statement using the \(\varepsilon ,\delta \) definition of a limit.
30. \(\mathop {\lim }\limits_{x \to 2} \left( {{x^2} + 2x - 7} \right) = 1\)
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