Limits and Continuity in Mathematics: Exploring Functions and Calculating Limits
This content delves into the concept of limits in mathematics, focusing on functions, limit laws, one-sided limits, and limits involving trigonometric functions. It includes examples and explanations on how to calculate limits using various techniques.
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Presentation Transcript
MATHEMATICSI FIRST SEMESTER LimitsAnd Continuity 11
Outlines Limit of a Function Calculating Limits Using the Limit Laws One-Sided Limits and Limits at Infinity One-Sided Limits Limits Involving (sin ?)/? Finite Limitsas ContinuousFunctions x
Limit of a Function Let (x) be defined on an open interval about x0 except possibly at itself. We say that the limit of (x) as x approaches x0 is the number L, andwrite
Calculating Limits Using the Limit Laws The next theorem tells how to calculate limits of functions
Solution : We cannot substitute because it makes the denominatorzero.
Limits Involving (sin) A central fact about (sin?) ?is that in radian measure its limit as ? 0 is 1. We can see this in Figure, EXAMPLE7: Show that, a. b.
x Finite Limits as
EXAMPLE 9: Find the points at which the function in Figure is continuous andthe points at which is discontinuous. Solution: The function is continuous at every point in itsdomain [0, 4] except at x = 1, x = 2, and x =4.
