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Magazine Problem 74
In Exercises 71-78, find \(\lim_{h \to 0}\ \dfrac{f(x+h)-f(x)}{h} \). \(f(x) = \sqrt{x-2}\)
Problem 74
WRITING Write a brief description of the meaning of the notation $$\lim_{x \to 5} f(x)=12$$.
Problem 75
In Exercises 71-78, find \(\lim_{h \to 0}\ \dfrac{f(x+h)-f(x)}{h} \). \(f(x) = x^2-3x\)
Problem 75
THINK ABOUT IT Use a graphing utility to graph the tangent function. What are \(\lim_{x \to 0} \textrm{tan}\ x\) and \(\lim_{x \to \pi/x} \textrm{tan}\ x\)? What can you say about the existence of the \(\lim_{x \to \pi/2} \textrm{tan}\ x\)?
Problem 76
TRUE OR FALSE? In Exercises 75 and 76, determine whether the statement is true or false. Justify your answer. A tangent line to a graph can intersect the graph only at the point of tangency.
Problem 76
In Exercises 71-78, find \(\lim_{h \to 0}\ \dfrac{f(x+h)-f(x)}{h} \). \(f(x) = 4-2x-x^2\)
Problem 77
In Exercises 71-78, find \(\lim_{h \to 0}\ \dfrac{f(x+h)-f(x)}{h} \). \(f(x) = \dfrac{1}{x+2}\)
Problem 77
WRITING Use a graphing utility to graph the function given by \(f(x) = \dfrac{x^2 - 3x - 10}{x-5}\). Use the trace feature to approximate \(\lim_{x \to 4} f(x)\). What do you think \(\lim_{x \to 5} f(x)\) equals? Is \(f\) defined at \(x=5\)? Does this affect the existence of the limit as \(x\) approaches \(5\)?
Problem 78
In Exercises 71-78, find \(\lim_{h \to 0}\ \dfrac{f(x+h)-f(x)}{h} \). \(f(x) = \dfrac{1}{x-1}\)
Problem 79
FREE-FALLING OBJECT In Exercises 79 and 80, use the position function \(s(t) = -16t^2 + 256\) which gives the height (in feet) of a free-falling object. The velocity at time \(t = a\) seconds is given by \(\lim_{t \to a} [s(a) - s(t)]/(a-t)\). Find the velocity when \(t=1\) second.
