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1.2 An Introduction to Limits

1.2 An Introduction to Limits. We have a point discontinuity at x = 1. What happens as from the left and from the right?. as x approaches 1. .75 .9 .99 .999 1 1.001 1.01 1.1. x f(x). 2.31 2.71 2.97 2.99. ?. 3.003 3.03 3.3. f(x) aproaches 3.

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1.2 An Introduction to Limits

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  1. 1.2 An Introduction to Limits

  2. We have a point discontinuity at x = 1. What happens as from the left and from the right? as x approaches 1 .75 .9 .99 .999 1 1.001 1.01 1.1 x f(x) 2.31 2.71 2.97 2.99 ? 3.003 3.03 3.3 f(x) aproaches 3

  3. Sometimes we can find a limit by just plugging in the number we are approaching. Ex. Find the limit.

  4. Ex. Evaluate the function at several points near x = 0 and use the results to find the limit. -.01 -.001 -.0001 0 .0001 .001 .01 x 1.9949 1.99995 1.9995 ? 2.0049 2.00005 2.00050 f(x) f(x) approached 2 f(x) approached 2

  5. Ex. Find the limit as x 2 where f(x) = What is the y-value as x approaches 2 from the left and from the right? The limit is 1 since f(x) = 1 from the left and from the right as x approaches 2. The value of f(2) is immaterial!!!

  6. 3 types of limits that fail to exist. • Behavior that differs from the left and from • the right. Ex. -1 1 the limit D.N.E. , since the limit from the left does not = the limit from the right.

  7. 2. Unbounded behavior Ex. Since f(x) the limit D.N.E.

  8. 3. Oscillating behavior (use calculator) As x 0, f(x) oscillates between –1 and 1, therefore the limit D.N.E. Limits D.N.E. when: • f(x) approaches a different number from the • right side of c than it approaches from the left side. 2. f(x) increases or decreases without bound as x approaches c. • f(x) oscillates between two fixed values as x • approaches c.

  9. A Formal Definition of a Limit If f(x) becomes arbitrarily close to a single number L as x approaches c from either side, then we say that the limit of f(x), as x approaches c, is L. In the figure to the left, let represent a small positive number. Then the phrase “f(x) becomes arbitrarily close to L” means that f(x) lies in the interval (L - , L + ) (c,L)

  10. Def. of a Limit Let f be a function defined on an open interval containing c and let L be a real number. The statement means that for each whenever

  11. Finding a for a given . Given the limit: find To find delta, we establish a connection between Thus, we choose

  12. Finding a for a given .

  13. Finding a for a given . For all x in the interval (1,3)

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