Chapter 1: Limits
Section 1.1: Naive Limits
Apply the technique of rationalizing the numerator to obtain limx→0hx, where hx=1+x−1x.
Evaluating hx at x=0 will clearly result in a "division by zero" error. Note also that the numerator also evaluates to zero at x=0. Hence, the function h is not defined at x=0. But even though it is a fraction, h is not a rational function because it is not a ratio of two polynomials. Thus, the Factor and Remainder theorem does not apply. Yet, there may still be an algebraic transformation that results in the cancellation of a common factor of x−0. Indeed, there is, and this is the device of rationalizing the numerator.
To rationalize the numerator, multiply the fraction by 1 in the form 1+x+11+x+1, where 1+x+1 is referred to as the conjugate of 1+x−1.
=1+x−1x ⋅ 1+x+11+x+1
Table 1.1.4(a) Rationalization of the numerator in hx
Because all five expressions on the right in Table 1.1.4(a) are equal to hx, the restriction that x≠0 applies to all. However, the last one, regarded in its own right, can indeed be evaluated at x=0, and the resulting value, namely, 1/2, is the required limit. In other words, by transforming hx by rationalizing its numerator, an expression results that extends the domain of h. The value of the extended function at x=0 tells the behavior of hx near x=0.
Unfortunately, if hx is multiplied by this form of 1, Maple would reduce the multiplier to 1 immediately, and hx would be multiplied by just 1. In Maple, the numerator and denominator of hx must be separately multiplied by the "conjugate" of its numerator, as per the following calculation.
Control-drag (or type) hx=…
Context Panel: Assign Function
hx=1+x−1x→assign as functionh
Type (or copy/paste) the numerator of hx
Using a space between, multiply by the conjugate.
Context Panel: Expand≻Expand
Type (or copy/paste) the transformed numerator.
Divide by the denominator of h and by the conjugate of the numerator.
Press the Enter key.
Context Panel: Evaluate at a Point≻x=0
→evaluate at point
Evaluation palette: Limit template
Context Panel: Evaluate and Display Inline
limx→0hx = 12
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