LLC LearningExpress - 501 Calculus Questions

CHAPTER 8 Limit Problems The notion of a limit is the single most important underlying concept on which the calculus is built. We can use the notion of a limit to describe the behavior of a function near a particular input when it is not defined at the input. We first review some basic definitions and properties of limits. Intuitive Definition: Let f be a function that is defined in the vicinity of a , but not necessarily at a . If it is the case that as , there exists a corresponding real number L such that , then we say f has limit L at a and write . Basic Principles of Limits Principle 1: A function need not be defined at x = a in order to have a limit at x = a .

Principle 2: The functional value at x = a is not relevant to the limit at x = a . Principle 3: Knowing the functional value at x = a is not sufficient to describe the functions behavior in a vicinity of x = a . Principle 4: When a function oscillates too wildly near x = a , then there is no limit at x = a . (More precisely, if there are two sequences of inputs that approach x = a for which the corresponding functional values approach different real numbers, then there is no limit at x = a .) Principle 5: If the functional values approach L as x approaches a from the left, but the functional values approach M ( L ) as x approaches a from the right, then there is no limit at x = a . Principle 5 leads to the following definition. Definition: f has a left-hand limit L at x = a if as x approaches a from the left, the corresponding functional values approach the real number L .

We write . (The notion of a right-hand limit is defined in a similar manner.) The notion of one-sided limits leads to the following useful characterization of limits. Theorem If (or at least one of them does not exist), then does not exist. If , then . The following rules enable us to compute the limits of various arithmetic combinations of functions. Arithmetic of Limits Let n and K be real numbers, and assume f and g are functions that have a limit at c .

Then, we have: Rule (Symbolically) , provided , provided the latter is defined. Rule (in Words) Limit of a constant is the constant. Limit of x as x goes to c is c . Limit of a constant times a function is the constant times the limit of the function. Limit of a sum (or difference) is the sum (or difference) of the limits. Limit of a product is the product of the limits.

L imit of a quotient is the quotient of the limits, provided the denominator doesnt go to zero. Limit of a function to a power is the power of the limit of the function. All of these rules hold for left- and right-hand limits as well. If, upon applying these properties, the result is / or / , you cannot apply them directly. Rather, some algebraic simplification must first occur. Then, reapply them.

Definition: A function f ( x ) is continuous at x = a if . Definition: f ( x ) approaches negative (or positive) infinity as x a if the corresponding functional values become unboundedly negative (or positive) as x approaches a . We write (or ). We also can interpret such limits in terms of left- and right-hand limits. In all such cases, we say f is unbounded and call x = a a vertical asymptote of f . Definition: f has a limit L as x approaches (or ) if the functional values can be made arbitrarily close to a single real number L for a sufficiently large x .

We write or and say the line y = L is a horizontal asymptote of f . Questions For Questions 186 through 192, use the following graph to evaluate the given quantity.

g (1) At which x -values in the interval [1,7] is g discontinuous? For Questions 193 through 205, compute the indicated limit. a. b. c. d. a. b. c. d. d.

The limit does not exist. a. / b. c. The limit does not exist. a. b. / c. / d. / d.

The limit does not exist. a. The limit does not exist. b. c. / a. b. c. d. a. b. c. d.

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