Chapter 1: Limits
Section 1.2: Precise Definition of a Limit
Use Definition 1 to verify limx→3x=3.
Type the equation fx=…
Context Panel: Assign Function
fx=x→assign as functionf
Figure 1.2.7(a) is an animation in which fx=x is graphed in black, and y=3 is graphed in blue.
The slider in the animation toolbar controls the value of ϵ. As the slider is moved past the first frame, red and green horizontal lines delineate an ϵ-band around y=3 and red and green vertical lines delineate the band 3−δL<x<3+δR.
The red and green horizontal lines are drawn at y=3±ϵ, respectively, and the red and green vertical lines are drawn at the corresponding x-coordinates x=f−13−ϵ and x=f−13+ϵ.
Figure 1.2.7(a) Animation illustrating Definition 1.2.1
Write the equation fa−δL=L−ϵ
Press the Enter key.
Context Panel: Solve≻Isolate Expression for≻δL
Context Panel: Simplify≻Simplify
→isolate for delta[L]
Write the equation fa+δR=L+ϵ
Press the Enter key.
Context Panel: Solve≻Isolate Expression for≻δR
→isolate for delta[R]
Clearly, δL<δR, but the choice δϵ=ϵ is simpler.
Figure 1.2.7(b) suggests that δL=23ϵ−ϵ2>ϵ for 0<ϵ<23−1≐2.46.
To establish this inequality analytically, compare the left- and right-sides via the ratio
which is greater than 1 for 0<ϵ<23−1.
Figure 1.2.7(b) Graph of δϵ and the line y=ϵ
To complete the proof, show that x−3<δ=ϵ ⇒ fx−3<ϵ. This is done in Table 1.2.1 by showing that f3+t ϵ−3<ϵ, where t<1.
=3+t ϵ−33+t ϵ+33+t ϵ+3
=t ϵ3+t ϵ+3
<|t ϵ1| = t ϵ<ϵ
Table 1.2.7(a) Verification that x=3+t ϵ ⇒ f3+t ϵ−3<ϵ
The key step is in the second equality, where the "numerator" is rationalized, resulting in the third equality. The first inequality follows from the observation that the sum of the square roots in the preceding equality is greater than 1, so replacing this denominator with 1 makes the denominator smaller, and thus, the fraction larger. The remaining two steps are the same as in Examples 1.2.5 and 1.2.6.
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