We can also conclude from the information in the table that f is not continuous at 2 or 3 or 4, because f(x) f(2) , f(x) f(3) , and f(x) f(4). ≠ ≠ ≠ The behaviors at = 2 and = 4 exhibit a hole in the graph, sometimes called a removable x x discontinuity, since the graph could be made continuous by changing the value of a single point. The behavior at = 3 is called a jump discontinuity, since the graph jumps between two values. x So which functions are continuous? It turns out pretty much every function you’ve studied is continuous where it is defined: polynomial, radical, rational, exponential, and logarithmic functions are all continuous where they are defined. Moreover, any combination of continuous functions is also continuous. This is helpful, because the definition of continuity says that for a continuous function, . That means for a continuous function, we can find the limit by direct substitution (evaluating the function) if the function is continuous at a. Example 4 Evaluate using continuity, if possible: a) b) b) a) The given function is polynomial, and is defined for all values of so we can find the limit x, by direct substitution: b) The given function is rational. It is not defined at = -3, but we are taking the limit as x x approaches 2, and the function is defined at that point, so we can use direct substitution: c) This function is not defined at = 2, and so is not continuous at = 2. We cannot use direct x x substitution. 2.2 Exercises 1. Use the graph to determine the following limits. (a) (b) (c) (d) 2. Use the graph to determine the following limits. (a) (b)
We can also conclude from the information in the table that f is not continuous at 2 or 3 or 4, because f(x) ≠ f(2) , f(x) ≠ f(3) , and f(x) ≠ f(4). The behaviors at x = 2 and x = 4 exhibit a hole in the graph, sometimes called a removable discontinuity, since the graph could be made continuous by changing the value of a single point. The behavior at x = 3 is called a jump discontinuity, since the graph jumps between two values. So which functions are continuous? It turns out pretty much every function you’ve studied is continuous where it is defined: polynomial, radical, rational, exponential, and logarithmic functions are all continuous where they are defined. Moreover, any combination of continuous functions is also continuous. This is helpful, because the definition of continuity says that for a continuous function, . That means for a continuous function, we can find the limit by direct substitution (evaluating the function) if the function is continuous at a. Example 4 Evaluate using continuity, if possible: a) b) b) a) The given function is polynomial, and is defined for all values of x, so we can find the limit by direct substitution: b) The given function is rational. It is not defined at x = -3, but we are taking the limit as x approaches 2, and the function is defined at that point, so we can use direct substitution: c) This function is not defined at x = 2, and so is not continuous at x = 2. We cannot use direct substitution. 2.2 Exercises 1. Use the graph to determine the following limits. (a) (b) (c) (d) 2. Use the graph to determine the following limits. (a) (b)
BusCalc_Chp2_wExercises.docx