Limits The Limit of a Function Definition: “The limit of f (x), as x approaches a, equals L”, denoted by Lxfax=→)(lim means that the values of f (x) become arbitrarily close to L when x is taken to be sufficiently close to a (on either side of the point x = a) but not equal to a. Note: The limit L, if it exists, depends only on how the function behaves near a, but not what happens at the point x = a itself. The function doesn’t even need to be defined at a for a limit to exist. ex. Consider the three functions 1)(+=xxf, 11)(2−−=xxxg , and =≠−−=1,01,11)(2xxxxxh Find the limit of each function, if it exists, as x approaches 1. (Hint: what do their graphs look like?)When a limit does not exist: ex. xxπsinlim0→ (This limit is an example where numerical data can be very misleading!!!) Note: Therefore, numerical data alone CANNOT be used conclusively to find the limit, or even to show that the limit exists. ex. xx1lim0→ ex. 22lim2−−→xxxOne-Sided Limits Previously the definition of limit requires that f (x) approaches the limiting value L as x approaches a from both sides of the point x = a. If we relax the definition a little, and allow x to approach a from only one side rather than from both sides, what we get are called one-sided limits. Definition: We denote Lxfax=−→)(lim as the left-hand limit of f (x) as x approaches a is L. (Reads, usually, “the limit of f (x) as x approaches a from the left”, or “from below”.) It means that the values of f (x) become arbitrarily close to L when x is taken to be sufficiently close to a, from the left side of a (i.e. the side where x < a) but not equal to a. Similarly, the right-hand limit is denoted by Lxfax=+→)(lim Reads, usually, “the limit of f (x) as x approaches a from the right”, or “from above,” is L. It means that the values of f (x) become arbitrarily close to L when x is taken to be sufficiently close to a, from the right side of a (i.e. the side where x > a) but not equal to a. Now just by comparing the definition of the regular (two-sided) limit with those of the one-sided limits, we see that, obviously: Theorem: Lxfax=→)(lim if and only if Lxfax=−→)(lim and Lxfax=+→)(lim That is, if the 2 one-sided limits exist and are equal to L, then the limit exists and is L, and vice versa.ex. Show that the limit does not exist. (2.3, #47) 11lim21−−→xxx The numerator factors into (x + 1)(x – 1), the denominator is a little subtle, however. When x approaches 1 from the right, the difference x – 1 is a small, but always positive, number, which is unchanged by taking the absolute value. The right-hand side limit is, therefore, 2)1(lim1)1)(1(lim11lim1121=+=−−+=−−+++→→→xxxxxxxxx Note: In the last step of the calculation above, the value x = 1 is substituted directly into the expression (x + 1) to obtain the limit of 2. This step is perfectly legitimate, at least in this case. We will discuss this method, called direct substitution, very shortly. When x approaches 1 from the left, however, the difference x – 1 is a small, but always negative, number, which is changed when its absolute value is taken. It becomes – (x – 1) instead. (Recall that the absolute value operation strips away the negative sign if the number is less than 0, which is equivalent to multiplying the negative number by an additional factor of –1 to cancel the negative sign.) With that in mind, the left-hand side limit is, therefore, 21211lim)1()1)(1(lim11lim1121−=−=−+=−−−+=−−−−−→→→xxxxxxxxx As a result, the right-hand and left-hand limits are, respectively, 2 and –2. Since they are different, therefore, the limit does not exist.Infinite Limits and Vertical Asymptotes Definition: Let f be a function defined on both sides of a, except possibly at a itself. Then ∞=→)(lim xfax means that the values of f (x) can be made arbitrarily large (i.e., it grows without bound) as x becomes sufficiently close to, but not equal to, a. Similarly, ∞−=→)(lim xfax means that the values of f (x) can be made arbitrarily large negative (i.e., it decreases without bound) as x becomes sufficiently close to, but not equal to, a. Note: In such cases it does NOT mean that the limit exists (recall that ∞ and −∞ are not even actual numbers). It is just a specific way to describe a limit that fails to exist due to the fact that f is increasing or decreasing without bound as x approaches a. ex. Let f (x) = 1 / x2, then ∞=→)(lim0xfx ex. Consider the function f (x) = 1 / x. It has two infinite one-sided limits: ∞−=−→)(lim0xfx and ∞=+→)(lim0xfx But the (two-sided) limit does not exist as x approaches 0 because the two one-sided limits are different (they grow unbounded in opposite directions). Similarly, xxtanlim2/π→ does not exist because the two one-sided limits are not equal.Definition: The line x = a is a vertical asymptote of the curve y = f (x) if any one of the following conditions is true: ∞=→)(lim xfax , ∞=−→)(lim xfax , ∞=+→)(lim xfax, ∞−=→)(lim xfax , ∞−=−→)(lim xfax , ∞−=+→)(lim xfax In short, x = a is a vertical asymptote if either one of the two one-sided limits of f (x) as x approaches a is ∞ or −∞. ex. f (x) = 1 / x and g(x) = 1 / x2 both have a vertical asymptote x = 0. ex. h(x) = tan x has vertical asymptotes at x = ±π/2, ±3π/2, ±5π/2, …Limit Laws Suppose c is any constant and that )(lim xfax→ and )(lim xgax→ exist. Then 1. )(lim)(lim)]()([lim xgxfxgxfaxaxax →→→+=+ 2. )(lim)(lim)]()([lim xgxfxgxfaxaxax →→→−=− 3. )(lim)(lim xfcxcfaxax →→= 4. )(lim)(lim)]()([lim xgxfxgxfaxaxax →→→⋅= 5. )(lim)(lim)()(limxgxfxgxfaxaxax→→→= if 0)(lim≠→xgax 6. naxnaxxfxf )](lim[)]([lim→→= n > 0 7. naxnaxxfxf )(lim)(lim→→= where n is a positive integer ex. Suppose 3)(lim5=→xfx and 7)(lim5=→xgx, then a. 2173)]()([lim5=⋅=→xgxfx b. 17)3(2)]()(2[lim5−=−=−→xgxfx c. 73)()(lim5−=−→xgxfxEvaluating a limit of polynomial or rational function First, two special limits of very intuitive nature: For any real numbers a and c, the limit of a constant function ccax=→lim the limit of f (x) = x axax=→lim As a