Big Idea 1: Limits
"It is this basic idea of a limit that sets calculus apart from other areas of mathematics. In fact, we could define calculus as the part of mathematics that deals with limits."
- James Stewart, Calculus, 2003
As mentioned above, limits are the foundation of calculus. Although limits themselves are rarely used for anything useful, all of our useful calculus ideas come from limits. Because of this, if you don't understand the idea of a limit, you're much more likely to use the formulas wrong, or not have a clue what you're really doing.
EU 1.1 Shows you everything you need to know about using limits
EU 1.1 Shows you how to use limits to determine if a function is continuous at a point
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Essential Understanding 1.1 Students will understand that the concept of a limit can be used to understand the behavior of functions.
Learning Objective 1.1A(a) Students will be able to express limits symbolically using correct notation and (b) Interpret limits expressed symbolically.
Essential Knowledge 1.1A1 Students will know that given a function f, the limit of f(x) as x approaches c is a real number R if f(x) can be made arbitrarily close to R by taking x sufficiently close to c (but not equal to c). If the limit exists and is a real number, then the common notation is
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Essential Knowledge 1.1A2 Students will know that the concept of a limit can be extended to include one-sided limits, limits at infinity, and infinite limits.
Essential Knowledge 1.1A3 Students will know that a limit might not exist for some functions at particular values of x. Some ways that the limit might not exist are if the function is unbounded, if the function is oscillating near this value, or if the limit from the left does not equal the limit from the right.
Learning Objective 1.1B Students will be able to estimate limits of functions.
Essential Knowledge 1.1B1 Students will know that numerical and graphical information can be used to estimate limits.
Learning Objective 1.1C Students will be able to determine limits of functions.
Essential Knowledge 1.1C1 Students will know that limits of sums, differences, products, quotients, and composite functions can be found using the basic theorems of limits and algebraic rules
Essential Knowledge 1.1C2 Students will know that the limit of a function may be found by using algebraic manipulation, alternate forms of trigonometric functions, or the squeeze theorem
Essential Knowledge 1.1C3 Students will know that limits of the indeterminate forms and may be evaluated using L'HĂ´pital's Rule
Learning Objective 1.1D Students will be able to deduce and interpret behavior of functions using limits.
Essential Knowledge 1.1D1 Students will know that asymptotic and unbounded behavior of functions can be explained and described using limits.
Essential Knowledge 1.1D2 Students will know that relative magnitudes of functions and their rates of change can be compared using limits.
Essential Understanding 1.2 Students will understand that continuity is a key property of functions that is defined using limits.
Learning Objective 1.2A Students will be able to analyze functions for intervals of continuity or points of discontinuity.
Essential Knowledge 1.2A1 Students will know that a function f is continuous at x = c provided that f(c) exists, exists, and
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Essential Knowledge 1.2A2 Students will know that polynomial, rational, power, exponential, logarithmic, and trigonometric functions are continuous at all points in their domains.
Essential Knowledge 1.2A3 Students will know that types of discontinuities include removable discontinuities, jump discontinuities, and discontinuities due to vertical asymptotes.
Learning Objective 1.2B Students will be able to determine the applicability of important calculus theorems using continuity.
Essential Knowledge 1.2B1 Students will know that continuity is an essential condition for theorems such as the Intermediate Value Theorem, the Extreme Value Theorem, and the Mean Value Theorem.