Limits Limit Laws Suppose c is constant, n is a positive integer, and f() and g() both eist. Then,. [f() + g()] = f() + g() 2. [f() g()] = f() g() [ ] 3. [c f()] = c f() [ ] [ ] 4. [f() g()] = f() g() f() 5. g() = 6. [f()] n = 7. c = c 8. = a 9. n = a n f() g(), as long as [ ] n f() g() 0. 0. n = n a. If n is even, then only true for a > 0.. n f() = n f(). If n is even, then only true when f() > 0. These laws lead to the following:. Direct Substitution Property If f() is a polynomial [remember that this includes constant and linear functions], rational, radical, or trigonometric function and a is in the domain of f (and not a zero of f if f is a radical function), then f() = f(a) (i.e., we can evaluate the it by simply plugging in = a). In general, direct substitution should work as long as nothing goes wrong at or around = a. Eamples of something going wrong would be vertical asymptotes, a function being undefined on one side of a, or changing to another piece of a piecewise-defined function.
2 Algebraic Techniques for Finding Limits First a useful theorem: Theorem. If f() = g() ecept at = a, then f() = g() (if the its eist). This theorem makes some of the algebraic manipulations we ll be doing legal. 2. Algebraic Techniques Here are some common algebraic techniques that arise when finding its: Canceling factors. (This is possible because of the theorem above.) Putting terms over a common denominator. Rationalizing, i.e., multiplying/dividing by a conjugate. Analyzing behavior under a radical. A combination of the above. 2.2 Finding Limits of ± If, after canceling as much as possible, we have f(a) = c, where c 0, then 0 f() = ± ± (As long as f() is a nice function, such as a rational or trig function.) To determine whether the it is + or, we need to determine whether the function is positive or negative near a (typically we ll need to check on the left and right of a separately). 2.3 Limits Involving Absolute Value Recall that a function involving absolute value can be epressed as a piecewise-defined function. For eample, {, if 0 =, if < 0 When evaluating the it of a function involving absolute value, we first write the function in piecewise form, then take left- and right-hand its (using whichever technique is appropriate). 2.4 Squeeze Theorem Squeeze Theorem. If f() g() h() when is near a, but not necessarily at a [for instance, g(a) may be undefined], and f() = h() = L, then g() = L also. 2
2.4. Eamples Eample. Find ( ) 2 cos 2. When trying to find functions to use to squeeze g(), we want functions that are, a) similar enough to g() that we can be sure the squeeze works, b) easier to evaluate their it as a. We typically do this by starting with the most complicated or troublesome part of g(), see if we can find constants (or simpler functions) that it stays between, and then multiply in the rest of nicer parts of g(). In this case, the part of g() that is giving us the most trouble is the cos ( ) part (we get division by 0 if 2 we try direct substitution). Now we know that cosine stays between - and, so ( ) cos 2 for any in the domain of the function (i.e., any 0). Since 2 is always positive, we can multiply this inequality through by 2 : ( ) 2 2 cos 2 2 So, our original function is bounded by 2 and 2. Now since then, by the Squeeze Theorem, Eample 2. Find 2 = 2 = 0, ( ) 2 cos 2 = 0. 2 e sin( ). As in the last eample, we have a problem arising from division by 0 inside the trig term. Now the range of sine is also [, ], so ( ) sin. Taking e raised to both sides of an inequality does not change the inequality, so e e sin( ) e, and, as in the last eample, we can multiply through by 2 and get 2 e 2 e sin( ) 2 e. So, our original function is bounded by e 2 and e 2, and since e 2 = e 2 = 0, then, by the Squeeze Theorem, 2 e sin( ) = 0. 3
2.5 Two Special Limits Eample 3. Find sin() cos() =, = 0 sec() sec() Direct substitution leads to division by 0. If we rewrite the secants as can write this as it as: sec() = sec() = cos() cos() cos() cos() cos() cosine cos() = cos() cos() cos() = = cos() ( ) cos() = = (0) = 0 s and do some rearranging, we (We could have also multiplied and divided the original equation by cos() and simplified. The it would been the same.) 2.6 Finding Limits at ± Definition. f() = L means that we can make f() arbitrarily close to L by taking sufficiently large positive values of. So, as grows larger, the function values approach L. Similarly, f() = L means that we can make f() arbitrarily close to L by taking sufficiently large negative values of. So, as grows larger in the negative direction, the function values approaches L. The line y = L is called a horizontal asymptote. We can also combine its at ± with the idea of its of ± and talk about f() = or or Talking about a large negative number is somewhat ambiguous, but we will take it to mean that the number is negative and far away from 0, i.e., the number is negative and its magnitude (absolute value) is large. 4
The following theorem is used when evaluating almost all infinite its: Theorem 2. Let c be a real number and let r be a positive rational number. Then c r = 0 If r is defined for all values of in the interval (, 0) then c r = 0 When taking the it as we can t have a value of r that is equivalent to taking an even root (e.g., r = 2 would cause us to be taking the square root of negative values). To evaluate its at ± we will almost always divide through by the largest power of in the denominator, simplify, and then use the result of the theorem to evaluate. While we can t really plug in = ± (since isn t a number), we can sometimes think that way and use the following sloppy notation to evaluate certain its at infinity. (Note: in all of the following, when we write what we really mean is something approaching [as approaches some value] ; similarly, really means something approaching [as approaches some value].), if c > 0 c = 0, if c = 0 (This is not the same as c approaching 0.), if c < 0, if c > 0 c = 0, if c = 0 (Again, this is not the same as c approaching 0.), if c < 0 = = = = + = = + c = for any real number c. + c = for any real number c. =?? We cannot say anything about this form! ± (something approaching 0) =?? We cannot say anything about this form, either! 5