Notes for 2.2 The Limit of a Function

2.2 The Limit of a Function

  • Introductory Example

    • Consider the function f(x) = x√x + 4 − 2. The table of values near x = 0 shows f(x) approaching 4 from both sides:
    • As x → 0−: f(x) ≈ 3.8708287, 3.9748418, 3.9874607, 3.999750, 3.999975, …
    • As x → 0+: f(x) ≈ 4.1213203, 4.0248457, 4.0124612, 4.0024984, 4.000250, 4.000025, …
    • Conclusion: lim_{x→0} f(x) = 4, i.e. the values of f(x) get arbitrarily close to 4 as x approaches 0 from either side.
    • Important note: f(0) is undefined in this example because the expression evaluates to 0/0 at x = 0 (the function value at 0 does not exist, even though the limit exists).
    • Key idea: When dealing with limits, we analyze f(x) as x approaches a, not necessarily at x = a. The limit may exist even if f(a) is undefined or f(a) differs from the limit value.
    • Comment on the different pictures: In all three pictures mentioned, lim_{x→4} f(x) = 2, showing that the limit can exist independently of the actual value of the function at the point (or even its existence).

  • Left-Handed and Right-Handed Limits

    • Left-handed limit: lim⁡x→a−f(x)\lim_{x\to a^-} f(x) is the limit when x approaches a from values less than a.
    • Right-handed limit: lim⁡x→a+f(x)\lim_{x\to a^+} f(x) is the limit when x approaches a from values greater than a.
    • Existence criterion: The limit lim⁡<em>x→af(x)=L\lim<em>{x\to a} f(x) = L exists if and only if both the left-handed and right-handed limits exist and are equal: lim⁡</em>x→af(x)=Lifflim⁡<em>x→a−f(x)=L and lim⁡</em>x→a+f(x)=L.\lim</em>{x\to a} f(x) = L \quad \text{iff} \quad \lim<em>{x\to a^-} f(x) = L \text{ and } \lim</em>{x\to a^+} f(x) = L.

  • Infinite Limits and Vertical Asymptotes (Introductory Concepts)

    • Infinite limits: Occur when f(x) grows without bound as x approaches a. Example: lim_{x→0} 1/x^2 = ∞.
    • As x → 0 from either side, 1/x^2 → ∞ since the denominator goes to 0 and the square keeps the sign positive.
    • Left- and right-hand infinite limits:
    • lim_{x→0^+} 1/x = ∞
    • lim_{x→0^-} 1/x = −∞
    • Definition: The line x = a is a vertical asymptote if the limit from the left, the limit from the right, or both are ±∞.
    • Example task (graph-based exercise): Given a graph of f, determine the indicated limits and identify vertical asymptotes. Typical limits to evaluate from the graph include:
    • lim_{x→4^-} f(x)
    • lim_{x→4^+} f(x)
    • lim_{x→4} f(x)
    • lim_{x→2^-} f(x)
    • lim_{x→2^+} f(x)
    • lim_{x→2} f(x)
    • lim_{x→−1} f(x)
    • lim_{x→−6} f(x)
    • lim_{x→−3^-} f(x)
    • lim_{x→−3^+} f(x)
    • lim_{x→−3} f(x)
    • lim_{x→6} f(x)
    • Question prompt from the graph: “What are the vertical asymptotes of f(x)?” (Values implied by the graph may indicate vertical lines where limits diverge to ±∞.)

  • Infinite and Vertical Asymptote Review from a Graph and Limits Perspective

    • Summary: If the limit as x approaches a is infinite (or does not exist due to divergence to ±∞) then x = a is a vertical asymptote (or the behavior indicates asymptotic behavior near a).

  • Horizontal vs. Vertical vs. Holes in Rational Functions (Conceptual Reminder)

    • Not every division by zero creates a vertical asymptote.
    • If a factor cancels in the rational expression, there is a hole at that x-value, not a vertical asymptote, because the division by zero is effectively resolved by cancellation.

  • Vertical Asymptotes of a Specific Function (Practice)

    • Function: f(x)=x−2x2−6x+8.f(x) = \frac{x - 2}{x^2 - 6x + 8}.
    • Factor the denominator: x2−6x+8=(x−2)(x−4).x^2 - 6x + 8 = (x - 2)(x - 4).
    • Simplify by canceling the common factor (for x ≠ 2):
      f(x)=x−2(x−2)(x−4)=1x−4.f(x) = \frac{x - 2}{(x - 2)(x - 4)} = \frac{1}{x - 4}.
    • Implications:
    • There is a hole at x = 2 (since the original expression is undefined there due to the canceled factor).
    • The simplified form shows a vertical asymptote at x = 4, since near x = 4 the function behaves like 1/(x − 4) which tends to ±∞.
    • Conclusion: Vertical asymptote at x = 4; hole at x = 2.

  • Connections and Takeaways

    • Limits describe behavior of f(x) as x approaches a, independent of f(a).
    • Left/right limits must agree for a two-sided limit to exist.
    • Infinite limits lead to vertical asymptotes; cancellation leads to holes rather than asymptotes.
    • In rational functions, factorization and cancellation help identify holes vs. vertical asymptotes.

  • Practical implications

    • When modeling real-world phenomena, vertical asymptotes indicate a variable blowing up to infinity at a certain point, signaling a breakdown of the model or a boundary where the model changes behavior.
    • If a discontinuity is a hole (removable discontinuity), it may be possible to redefine the function at that point to restore continuity except for the original domain constraints.

  • Formulas to remember (in LaTeX)

    • Left-hand limit: lim⁡<em>x→a−f(x)\lim<em>{x\to a^-} f(x) and Right-hand limit: lim⁡</em>x→a+f(x)\lim</em>{x\to a^+} f(x)
    • Two-sided limit: lim⁡<em>x→af(x)=L\lim<em>{x\to a} f(x) = L exists iff both one-sided limits exist and equal L: lim⁡</em>x→af(x)=L  ⟺  lim⁡<em>x→a−f(x)=L  &  lim⁡</em>x→a+f(x)=L.\lim</em>{x\to a} f(x) = L \iff \lim<em>{x\to a^-} f(x) = L \;\&\; \lim</em>{x\to a^+} f(x) = L.
    • Infinite limit example: lim⁡x→01x2=∞.\lim_{x\to 0} \frac{1}{x^2} = \infty.
    • Vertical asymptote definition: line x = a is a vertical asymptote if the corresponding one-sided limits are infinite.
    • Rational function cancellation example: If f(x)=p(x)q(x)f(x) = \frac{p(x)}{q(x)} and q(x)=(x−a) r(x)q(x) = (x - a)\,r(x) and p(a) = 0 as well so that a factor cancels, there is a hole at x = a, not necessarily a vertical asymptote.

  • Title for the notes

    • Notes for 2.2 The Limit of a Function (study guide)