Notes on Left-hand Limits, Power Rules, and Linearity of Limits (Transcript-Based)

Left-hand limit intuition at x → 2

• The speaker describes approaching a point from the left (lower x-values) and asks where the function value lands on the graph (the wall).
• Example interpretation: if you approach x = 2 from the left, you might have a left-hand limit of the function, which the speaker states equals 3. This suggests the concept of a left-hand limit:
  • If $\lim_{x \to 2^-} f(x) = 3$, then the function values from the left approach 3.
  • The same idea applies to any target point a: approaching from the left means x → a⁻.
• Takeaway: left-hand limits are the values that f(x) approaches as x gets arbitrarily close to a from below, and they may or may not equal the two-sided limit depending on continuity at a.

Power rules with limits

• If you have a function raised to a power, you can often pass the limit through the exponent:
  • For integer powers: if $\lim<em>{x \to a} f(x) = L$ and n is an integer, then $\lim</em>{x \to a} [f(x)]^{n} = L^{n}.$
  • Intuition: you can take the limit of the base first, then apply the power, since exponentiation is a continuous operation for real bases when the base is defined.
• For fractional powers (real exponents) there are extra conditions:
  • If the exponent is a rational number with denominator q (i.e., exponent = p/q), and you want to pass the limit inside the fractional power, you typically need the base to stay nonnegative near the limit (so the real-valued result is well-defined):
    $\lim<em>{x \to a} [f(x)]^{p/q} = \left(\lim</em>{x \to a} f(x)\right)^{p/q}, \quad \text{provided } f(x) \ge 0 \text{ near } a.$
  • If f(x) crosses negative values near the limit and the exponent is not an integer, the expression may be undefined in the reals, which invalidates a straightforward limit pass-through. In such cases, one must restrict the domain to where the expression is defined or work in complex numbers.
• Practical takeaway: for well-defined limits with powers, you can often interchange limit and exponent, but be mindful of domain when fractional powers are involved.

Linearity of limits (scalar multiplication)

• If a function is multiplied by a constant, you can pull the constant out of the limit:
  • If c is a constant and $\lim<em>{x \to a} f(x) = L$, then $\lim</em>{x \to a} [c \cdot f(x)] = c \cdot L.$
• This is a direct consequence of the limit laws and reflects the linearity of limits.
• Example: if $\lim<em>{x \to a} f(x) = 5$ and c = 7, then $\lim</em>{x \to a} [7 \cdot f(x)] = 7 \cdot 5 = 35.$

Special case: exponent equal to zero

• When the exponent is identically zero (i.e., the expression is [f(x)]^0), the value is 1 wherever the expression is defined:
  • For all x where f(x) ≠ 0, $[f(x)]^{0} = 1.$
  • If you take the limit as x → a, and the function is defined for x near a with f(x) ≠ 0, then
    $\lim_{x \to a} [f(x)]^{0} = 1.$
• Note: issues can arise if f(x) = 0 for x in a neighborhood, since 0^0 is indeterminate in some contexts. In limit problems, the common resolution is that the expression is treated by the rule above wherever it is defined, and the limit tends to 1 if the function values keep yielding a constant power of zero.
• A concrete illustration: if $\lim_{x \to a} f(x) = L$ with L > 0, then near a we can often treat $[f(x)]^{0} = 1$, so the limit is simply 1.

Putting it together: a small worked scenario

• Suppose you know from the left that $\lim_{x \to 2^-} f(x) = 3.$ Then:
  • Power rule (integer): $\lim_{x \to 2^-} [f(x)]^{2} = 3^{2} = 9.$
  • Fractional power (e.g., square root): if f(x) stays nonnegative near 2, $\lim_{x \to 2^-} \sqrt{f(x)} = \sqrt{3}.$ If f(x) takes negative values near 2, the real-valued square root may be undefined, and the limit analysis would need adjustment.
  • Scalar multiplication: if you consider $\lim<em>{x \to 2^-} [7 f(x)] = 7 \cdot \lim</em>{x \to 2^-} f(x) = 7 \cdot 3 = 21.$
  • If the exponent is zero: $\lim_{x \to 2^-} [f(x)]^{0} = 1.$ (assuming f(x) ≠ 0 near 2.)

Final step: plugging in the limit value

• In many problems, after applying the limit laws, you substitute the limit value to obtain a numerical result.
  • Example continuation: if after simplification you arrive at an expression whose limit requires substituting x = 9, you would compute the value at that point, provided the expression remains defined there.
• Practical workflow:
  • Identify the limit point a and one-sided nature if specified (left, right, or two-sided).
  • Determine the limit of the base function f(x) as x approaches a (and ensure the limit exists).
  • Apply the appropriate limit law for powers and composition, noting any domain restrictions (especially for fractional powers).
  • Apply linearity and constant multiplication as needed.
  • Evaluate the resulting expression by substitution of the limit value, ensuring the substituted value does not lead to undefined operations (like 0^0 or division by zero).

Connections to broader concepts

• These limit laws underpin continuity: if f is continuous at a, then lim_{x→a} f(x) = f(a), and you can freely pass limits through algebraic operations on f.
• They also underpin differentiation via the limit definitions, where evaluating limits of composed functions is essential.
• Real-world relevance: in modeling, limits allow you to understand behavior near critical points (e.g., thresholds, singularities) and to justify exchanging limiting processes with algebraic manipulations.

Summary of key rules (compact reference)

• Left-hand limit intuition: if $\lim_{x \to a^-} f(x) = L,$ then the left-hand limit of any operation on f behaves predictably by applying the operation to L, with caveats for domain.
• Power rule: $\lim<em>{x \to a} [f(x)]^{n} = \, [\lim</em>{x \to a} f(x)]^{n}$ for integer n; for fractional n = p/q, require nonnegativity of f near a (or appropriate domain handling).
• Linearity: $\lim<em>{x \to a} [c \cdot f(x)] = c \cdot \lim</em>{x \to a} f(x)$ for constant c.
• Zero exponent caveat: if the exponent is identically zero, the expression tends to 1 wherever defined; special care around points where the base is zero.

Note: The concepts above are foundational for manipulating limits in calculus, and the specific numerical examples (e.g., left-hand limit of 3 at x = 2, subsequent squaring yielding 9, etc.) illustrate how the laws are applied in practice.