Chapter 2: Limits and Derivatives

The graphs of \(f\) and \(g\) are given. Use them to evaluate each

limit, if it exists. If the limit does not exist, explain why.

(a) \(\mathop {lim}\limits_{x \to 2} \left( {f\left( x \right) + g\left( x \right)} \right)\)

(b) \(\mathop {lim}\limits_{x \to 0} \left( {f\left( x \right) - g\left( x \right)} \right)\)

(c) \(\mathop {lim}\limits_{x \to - 1} \left( {f\left( x \right)g\left( x \right)} \right)\)

(d) \(\mathop {lim}\limits_{x \to 3} \frac{{f\left( x \right)}}{{g\left( x \right)}}\)

(e) \(\mathop {lim}\limits_{x \to 2} \left( {{x^2}f\left( x \right)} \right)\)

(f) \(f\left( { - {\bf{1}}} \right) + \mathop {lim}\limits_{x \to - {\bf{1}}} g\left( x \right)\)

2. (a) Can the graph of \(y = f\left( x \right)\) intersect a vertical asymptote? Can it intersect a horizontal asymptote? Illustrate by sketching graphs.

(b) How many horizontal asymptotes can the graph of \(y = f\left( x \right)\) have? Sketch graphs to illustrate the possibilities.

Explain what it means to say that

\(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{1}}^ - }} f\left( x \right) = {\bf{3}}\)and \(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{1}}^ + }} f\left( x \right) = {\bf{7}}\)

In this situation, is it possible that\(\mathop {{\bf{lim}}}\limits_{x \to {\bf{1}}} f\left( x \right)\) exists? Explain.

Use the given graph to estimate the value of each derivative. Then sketch the graph of \(f'\).

(e) \(f'\left( 1 \right)\) (f) \(f'\left( 2 \right)\) (g) \(f'\left( 3 \right)\)

Find the derivative of the function

\(f\left( x \right) = \frac{{{x^{\bf{4}}} - {\bf{5}}{x^{\bf{3}}} + \sqrt x }}{{{x^{\bf{2}}}}}\)

in two ways: by using the Quotient Rule and by simplifying first.Show that your answers are equivalent. Which method do you prefer?

2: If \(f\) is continuous on \(\left( { - \infty ,\infty } \right)\), what can you say about its

graph?

Graph the curve\(y = {e^x}\)in the viewing rectangles\(\left( {{\bf{ - 1,1}}} \right)\)by\(\left( {{\bf{0,2}}} \right)\), \(\left( {{\bf{ - 0}}{\bf{.5,0}}{\bf{.5}}} \right)\)by\(\left( {{\bf{0}}{\bf{.5,1}}{\bf{.5}}} \right)\), and\(\left( {{\bf{ - 0}}{\bf{.1,0}}{\bf{.1}}} \right)\)by\(\left( {{\bf{0}}{\bf{.9,1}}{\bf{.1}}} \right)\). What do you notice about the curve as you zoom in toward the point\(\left( {{\bf{0,1}}} \right)\)?

Use the given graph of \(f\) to find a number \(\delta \) such that if \(0 < \left| {x - 3} \right| < d\), then \(\left| {f\left( x \right) - 2} \right| < 0.5\)

11-34: Evaluate the limit, if it exists.

30. \(\mathop {\lim }\limits_{x \to 2} \frac{{{x^2} - 4x + 4}}{{{x^4} - 3{x^2} - 4}}\)

If \(g\left( x \right) = {x^4} - 2\), find \(g'\left( 1 \right)\) and use it to find an equation of the tangent line to the curve \(y = {x^4} - 2\) at the point \(\left( {1, - 1} \right)\).