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MA103 Lab Report 2 Prep Page 1 MA103 Lab Report 2 Prep – Limits Limits at a Number (Text: 2.2, 2.3) To determine lim x a f ( x ) we are asking the question: As x gets closer and closer to the finite number a , which single value, if any, does f ( x ) approach? We can also consider one-sided limits: lim x a - f ( x ) [ lim x a + f ( x ) ] indicates the limit of f ( x ) as x approaches a from the left [ right ]. Often, we use the fact that lim x a f ( x ) = L if and only if lim x a - f ( x ) = lim x a + f ( x ) = L . To evaluate the limit of a function at the point a , direct substitution may be used if the function is continuous at a , and the substitution produces a determinate form. If direct substitution into a rational results in both a zero numerator and a zero denominator [ 0 0 , an indeterminate form ], then algebraic techniques may often be used to express the function in a determinate form. Such methods include: (i) factoring and cancelling; (ii) rationalizing [ either the numerator or denominator ]; (iii) finding common denominators; (iv) combinations of (i) – (iii) with simplification Example: lim x 16 4 - x 16 - x = lim x 16 4 - x 16 - x 4 + x 4 + x = lim x 16 16 - 4 x + 4 x - x (16 - x )(4 + x ) = lim x 16 16 - x (16 - x )(4 + x ) = lim x 16 1 (4 + x ) = 1 8 Note that the forms 0 · ∞ and ∞ - ∞ are also indeterminate and may require the use of algebra as well. In the case of an absolute-value or piecewise-defined function, left and right limits must be compared if a is a corner or a breaking point of the function, respectively. Infinite Limits and Vertical Asymptotes (Text: 2.2) For rational expressions, if lim x a g ( x ) h ( x ) yields the form 0 L [ where L is a finite non-zero number ], the limit has value 0. If it yields the form L 0 , the limit may approach or -∞ ; the sign, positive or negative, is determined by considering the sign of both the numerator and the denominator.

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