Chapter 2: Q34E (page 77)
15–42: Find the limit or show that it does not exist.
34. \(\mathop {{\rm{lim}}}\limits_{x \to \infty } \left( {{e^{ - x}} + 2{\rm{cos}}3x} \right)\)
The limit \(\mathop {{\rm{lim}}}\limits_{x \to \infty } \left( {{e^{ - x}} + 2{\rm{cos}}3x} \right)\) does not exist.
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If an equation of the tangent line to the curve \(y = f\left( x \right)\) at the point where \(a = {\bf{2}}\) is \(y = {\bf{4}}x - {\bf{5}}\), find \(f\left( {\bf{2}} \right)\) and \(f'\left( {\bf{2}} \right)\).
\({x^{\rm{2}}} + {y^{\rm{2}}} = {r^{\rm{2}}}\), \(ax + by = {\rm{0}}\).
Explain in your own words what is meant by the equation
\(\mathop {{\bf{lim}}}\limits_{x \to {\bf{2}}} f\left( x \right) = {\bf{5}}\)
Is it possible for this statement to be true and yet \(f\left( {\bf{2}} \right) = {\bf{3}}\)? Explain.
If \(f\left( x \right) = 3{x^2} - {x^3}\), find \(f'\left( 1 \right)\) and use it to find an equation of the tangent line to the curve \(y = 3{x^2} - {x^3}\) at the point \(\left( {1,2} \right)\).
Describe the intervals on which each function f is continuous.
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