Limits. What is a Limit? Limits are the spine that holds the rest of the Calculus skeleton upright. Basically, a limit is a value that tells you what.

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Presentation transcript:

Limits

What is a Limit? Limits are the spine that holds the rest of the Calculus skeleton upright. Basically, a limit is a value that tells you what height (y-value) a function is headed for or intended for, as you get close to or approach a specific x- value. It describes the behavior of the function as it gets closer to a particular value of x.

Formal Definition of a Limit For a fnc., f(x), and a real number, c, the limit of f(x) exists if and only if: 1.lim f(x) exists. There must be a limit from the left. 2.lim f(x) exists. There must be a limit from the right. 3.lim f(x) = lim f(x). The limit from the left must equal the limit from the right. (Note: this does not apply to limits as x approaches infinity!)

Evaluating Limits Look for a pattern in a series or a table Simple substitution Use Algebra and/or graph

Evaluate the following limits: 1. ½, 2/3, ¾, 4/5, 5/6, 6/7…….. 2. lim (3x-1) 3. lim 4. lim 5.lim

Evaluating One-Sided Limits 1. lim 2. lim 3. lim 4. lim

Evaluating Infinite Limits In general, a fractional function will have an infinite limit if the limit of the denominator is zero AND the limit of the numerator is NOT zero. The sign (+/-) of the infinite limit is determined by the sign of the quotient of the numerator and the denominator at values close to the number it is approaching. NOTE: a limit of is actually an indication that a real number limit does not exist (DNE) since it doesn’t make sense to say that the limit is infinitely unlimited!

Evaluate the following Limits: 1. lim 2. lim 3. lim

Limits at Infinity If the degree of the numerator and denominator are the same, then there is a horizontal asymptote and a limit at the coefficient of the numerator over the coefficient of the denominator. If the degree of the numerator is larger, then there is a limit at. If the degree of the denominator is larger, then there is a limit at 0.

Evaluate the following Limits: 1. lim 2. lim 3. lim 4. lim x 3 – x 2 – 3x

Limits involving Trig Fncs. lim sin x = sin c lim cos x = cos c lim

1. lim 2. lim cot (x) 3. lim 4. lim 5. lim