Numerical Estimation of Limits: Left/Right Approach and Examples

Numerical Method for Evaluating Limits

  • Definition: The limit of f(x) as x approaches a is a value l if the y-values get arbitrarily close to l as x approaches a from both sides. This is written as

l=lim⁡x→af(x)l = \lim_{x \to a} f(x)

  • One practical way to determine limits numerically is to sample x-values that get arbitrarily close to a from the left (x < a) and from the right (x > a), evaluate f(x), and compare the resulting y-values. If the left-hand and right-hand values tend to the same number, the limit exists and equals that number. If they don\'t, the limit does not exist.

  • Left-hand limit and right-hand limit notations:

lim⁡<em>x→a−f(x)=L</em>−,lim⁡<em> x→a+f(x)=L</em>+.\lim<em>{x \to a^-} f(x) = L</em>-, \quad \lim<em>{\,x \to a^+} f(x) = L</em>+.

  • The limit exists iff $L- = L+$ = L (finite). If either side fails to exist or if $L- \neq L+$, then $\lim_{x \to a} f(x)$ does not exist.

Example 1: Numerical limit for f(x) as x -> 1

Let

f(x)=5x2−7x+2x−1.f(x) = \frac{5x^2 - 7x + 2}{x - 1}.

We want lim⁡x→1f(x).\lim_{x \to 1} f(x).

  • Numerical data table (values of x approaching 1 from left and right):

    • Left side: x = 0.99, 0.999, 0.9999, …
    • Right side: x = 1.01, 1.0001, …

  • Example computed values (from the transcript):

    • f(0.99) = 2.95
    • f(0.999) = 2.995
    • f(1.0001) = 3.0005
    • (When these x-values are even closer to 1, the y-values get even closer to 3.)

  • Observation: As x approaches 1 from both sides, f(x) → 3.

  • Conclusion:

lim⁡x→1f(x)=3.\displaystyle \lim_{x \to 1} f(x) = 3.

  • Calculator tips (graphing routines):

    • Enter the simplified or factored form in Y =, e.g. using parentheses: (\frac{5x^2 - 7x + 2}{x - 1}).
    • If you cancel common factors, you can graph and use the Trace feature to read off y-values.
    • The difference between a direct table and the Trace method: a data table requires a constant spacing between x-values; Trace is often quicker for close-interval sampling.

  • Conceptual takeaway: A limit exists when the approach from both sides yields the same y-value; here both sides converge to 3.

Example 2: The limit of cos(1/x) as x -> 0

Consider

lim⁡x→0cos⁡(1x).\lim_{x \to 0} \cos\left(\frac{1}{x}\right).

  • Graphically, this is the classic oscillating function that does not settle down near 0.

  • Numerical sampling near 0 (values of x approaching 0 from left and right).

    • x = -0.01 and x = 0.01: f(x) ≈ 0.862
    • Choosing closer values: x = -0.001 and x = 0.001: f(x) ≈ 0.562
    • Even closer: f-values can jump to ≈ -0.952, and continue to vary as x gets smaller in magnitude.

  • Key conclusion: The y-values do not approach a single value as x -> 0; they oscillate with increasing frequency, so the limit does not exist.

  • Formal statement:

lim⁡x→0cos⁡(1x) does not exist.\lim_{x \to 0} \cos\left(\frac{1}{x}\right) \text{ does not exist}.

  • Note: The numerical method aligns with the graphical observation: no unique limiting value.

Example 3: Numerical analysis of a piecewise function to locate where the limit exists

  • Task: Find the values of c for which the limit

lim⁡x→cf(x)\displaystyle \lim_{x \to c} f(x)

exists for the given piecewise function (the exact formulas are not displayed in this transcript). The discussion focuses on c = 2 and c = 4.

  • Method: Evaluate f(x) for x-values approaching c from the left and from the right using the appropriate piece of the function.

  • For c = 2:

    • Approaching from the left uses the left-hand piece (e.g., the x^2-related expression)
    • Approaching from the right uses the right-hand piece (the piece on the other side of 2)
    • The computed y-values from both sides approach the same value: 4.
    • Conclusion: The limit exists at x = 2 and equals 4.

  • For c = 4:

    • Approaching from the left (the middle piece) yields values approaching 0.
    • Approaching from the right (the last piece) yields values approaching 4.
    • Since the left-hand and right-hand limits differ, the limit at x = 4 does not exist.

  • Overall conclusion: The limit exists for all real c except c = 4, as suggested by the graphical analysis (the graph showed a gap at that point).

  • Final note: The numerical method confirms the graphical method's conclusion for this example as well.

Takeaways and practical tips

  • Limit existence criterion: lim⁡<em>x→a−f(x)=lim⁡</em>x→a+f(x)=l(finite)⇒lim⁡x→af(x)=l.\lim<em>{x \to a^-} f(x) = \lim</em>{x \to a^+} f(x) = l \quad (\text{finite}) \Rightarrow \lim_{x \to a} f(x) = l. If not, the limit does not exist.

  • Numerical procedure:

    • Pick values of x on both sides of a that get arbitrarily close to a.
    • Evaluate f(x) and observe whether the y-values converge to the same value.
    • Use a calculator's Y= graphing, Trace, or a data table with consistent spacing to facilitate the checks.

  • Comparison with graphical method: The numerical approach can confirm what is seen graphically; it provides concrete y-values near the target a.

  • Practical caution: When using cos(1/x) near 0, expect no limit due to oscillation; do not misinterpret a few sample values as indicating a limit exists.

  • Reminder: Bring questions to class for further discussion.