OBJECTIVE 2(a): Determine whether a function is continuous at a point.

Level I

1. Define continuity in terms of limits. Compute limits of functions known to be continuous by evaluating at a limit point and not by some other process.

2. Formulate/select a limit statement that reveals the continuity status of a given function at a point.

3. Identify points of discontinuity from a table of values, a graph, and/or a (piecewise) algebraic description of a function.

Level II

1. Identify types of discontinuities from tabular data, a graph, and/or an algebraic description of a function:
   1. jump discontinuity
   2. removable discontinuity
   3. oscillatory behavior

2. Give examples of functions that are defined at a point but are discontinuous at that point. Describe such examples by tabular data, graphs, and/or algebraic definitions. Include suitable limit statements to establish the claim and characterize the type of discontinuity.

3. Describe the following characteristic features of the graphs of elementary functions using limit statements.
   1. They are continuous on their domains.
   2. Describe asymptotic behavior of exponential, logarithmic, and trigonometric functions.
   3. Non constant polynomials don't have horizontal or vertical asymptotes; their infinite limits at infinity are determined by the parity of the degree and the sign of the leading coefficient.

Level III

1. Identify discontinuities of functions that fail the Intermediate Value Theorem (IVT).

2. Apply the Intermediate Zero Theorem to conclude the existence of a given function on a given interval. Clearly identify continuity as a necessary hypothesis in this context.

3. Explain why the IVT applies or does not apply in a given situation by checking the hypotheses and conclusion against those of the IVT.

4. Determine whether a specifically given elementary function admits a continuous extension to boundary points of the domain (e.g. sin(x)/x does; |x|/x and sin(1/x) do not).

5. Use uniqueness of limits to argue that a given one-sided limit such as the limit as x-->a- of f(x) does not exist by exhibiting two sequences approaching (x_n) and (y_m) that approach a from the left such that f(x_n) and f(y_m) tend to distinct limits. (Example: f(x) = sin(1/x), x_n=2/((4n+1)Pi), y_m=1/(mPi))

Objective 2(b): Compute limits of elementary functions using limit laws and continuity.

Level I

1. Recognize a limit problem as a limit statement for which the limit is to be computed.

2. Apply the "algebraic limit laws" (additivity, scalar multiplication, multiplicative, quotient, root, and substitution) to compute limits of algebraic combinations of unknown functions with known limits.

Level II

1. Use the algebraic limit laws to compute specific limits.

2. Apply the continuity of the standard transcendental functions to compute specific limits of algebraic combinations of such functions.

Level III

1. Use the algebraic limit laws to argue that algebraic functions are continuous on their domains.

2. Apply comparison limit laws to compute specific limits. This includes the traditional squeeze law, as well as comparison tests designed to identify infinite limits and determinate forms of the type "bounded/infinity."

Objective 2(c): Compute limits of selected indeterminate forms, especially those arising from difference quotients of elementary functions.

Level I

1. Recognize indeterminate forms as limit problems that can not be solved directly using the limit laws (algebraic limit laws or comparison limit laws). Identify the following types of indeterminate forms:
   1. 0/0
   2. 1^infinity
   3. 0*infinity
   4. infinity - infinity

2. (Example: Suppose that lim as x-->1 of f(x) is 0, lim as x-->1 of g(x) is 0, lim as x-->1 of h(x) is +infinity, lim as x-->1 of i(x) = -infinity. Which of the following conclusions are valid: (a) lim as x-->1 of (f+g)(x) = 0 (b) lim as x-->1 of (f*h)(x) = 0 (c) lim as x-->1 of (f+h)(x) = +infinity (d) lim as x-->1 of (h+i)(x) = 0)

3. Identify direct computation of the derivative of a function as an indeterminate form of type 0/0.

Level II

1. Perform algebraic manipulations to enable the computation of certain indeterminate forms arising from limits of difference quotients of elementary functions. (Eg. binomial expansion, root conjugate trick, angle addition formulas.)

2. Compute variations of standard limits including the following.
   1. sinh/h -->1 as h --> 0
   2. (a^h - 1)/h --> ln(a) as h --> 0
   3. (1+1/n)^n --> e as n --> infinity

3. Relate the first two limits above to specific derivatives. Relate the last one to continuously compounded interest and the number e.

Level III

1. Compute limits of difference quotients by the following indirect method.
   1. Recognize the limit as the derivative of a certain function at a point.
   2. Compute the derivative of the function.
   3. Evaluate the derivative at a point to obtain the limit.

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