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Limits

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.

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Limits

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  1. Limits

  2. 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.

  3. 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: • lim f(x) exists. There must be a limit from the left. • lim f(x) exists. There must be a limit from the right. • 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!)

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

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

  6. Evaluating One-Sided Limits • lim • lim • lim • lim

  7. 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!

  8. Evaluate the following Limits: • lim • lim • lim

  9. 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.

  10. Evaluate the following Limits: • lim • lim • lim • lim x3– x2 – 3x

  11. Limits involving Trig Fncs. • lim sin x = sin c • limcos x = cos c • lim • lim

  12. lim • lim cot (x) • lim • lim • lim

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