121616949-math.47 - x → 1 x 3 = 4 Another of the most...

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2.3 Limits 33 It is worth commenting on the trivial limit lim x 1 5. From one point of view this might seem meaningless, as the number 5 can’t “approach” any value, since it is simply a fixed number. But 5 can, and should, be interpreted here as the function that has value 5 everywhere, f ( x ) = 5, with graph a horizontal line. From this point of view it makes sense to ask what happens to the height of the function as x approaches 1. Of course, as we’ve already seen, we’re primarily interested in limits that aren’t so easy, namely, limits in which a denominator approaches zero. There are a handful of algebraic tricks that work on many of these limits. EXAMPLE 2.9 Compute lim x 1 x 2 + 2 x - 3 x - 1 . We can’t simply plug in x = 1 because that makes the denominator zero. We can note that if x = 1, lim x 1 x 2 + 2 x - 3 x - 1 = lim x 1 ( x - 1)( x + 3)
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Unformatted text preview: x → 1 ( x + 3) = 4 Another of the most common algebraic tricks was used in section 2.1 . Here’s another example: EXAMPLE 2.10 Compute lim x →-1 √ x + 5-2 x + 1 . lim x →-1 √ x + 5-2 x + 1 = lim x →-1 √ x + 5-2 x + 1 √ x + 5 + 2 √ x + 5 + 2 = lim x →-1 x + 5-4 ( x + 1)( √ x + 5 + 2) = lim x →-1 x + 1 ( x + 1)( √ x + 5 + 2) = lim x →-1 1 √ x + 5 + 2 = 1 4 Occasionally we will need a slightly modified version of the limit definition. Consider the function f ( x ) = √ 1-x 2 , the upper half of the unit circle. What can we say about lim x → 1 f ( x )? It is apparent from the graph of this familiar function that as x gets close to 1 from the left, the value of f ( x ) gets close to zero. It does not even make sense to...
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