Essential Understanding 1.1

Essential Understanding 1.1 Students will understand that the concept of a limit can be used to understand the behavior of functions.

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What is a limit?

Basically, a limit is a way to get a x very close to a number without actually using that number. To clarify, means we want to see what happens if x gets really close to 5, but not actually become 5. For this I would think, well, so what would happen if x was like 4.9999999999 or 5.000000000001? Of course, in this case nothing special, x would basically be 5, and indeed 5 is the answer. This is a pretty boring example, but this is the basic idea. So what's the point of limits? Well sometimes interesting things do happen when x gets very close to a number without actually becoming that number.

Examples of the uses of limits

I won't spend too much time on this now, but basically limits are useful when 0 and ∞ mess up the normal math. For example:

Normal Math: x = ∞

for x = ∞ would give you , which is indeterminate and therefore kinda useless.

Limit Math: Limit as x approaches ∞

However, if we use limits instead, we can use some special limit rules that allow us to find the true answer

= = = 0

And 0 is a much more useful answer than indeterminate.

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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

.

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'Hôpital'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.