Therefore, limₓ→2 g(x) = 2 + 2 = 4. - Coaching Toolbox

Understanding the Context

What Does limₓ→2 g(x) = 4 Mean?

The notation limₓ→2 g(x) = 4 describes the value that the function g(x) approaches as x gets arbitrarily close to 2, but not exactly at x = 2. The limit exists and equals 4 if:

• g(x) produces output values approaching 4
  
• Every sequence of x values approaching 2 (but not equal to 2) results in g(x) values approaching 4
  
• The left-hand and right-hand limits both converge to the same number, ensuring continuity from both sides.

This means that even if g(2) is undefined, undefined or otherwise valued, the limit itself—what g(x) tends toward—remains firmly at 4.

Key Insights

How to Evaluate limₓ→2 g(x) = 4: Key Techniques

Determining that the limit is 4 typically involves algebraic manipulation, factoring, or application of known limit laws:

• Direct Substitution: First, always test x = 2 directly. If g(2) is finite and equals 4, the limit often equals 4 — provided continuity holds.
  
• Factoring and Simplification: If g(x) is a rational function (a ratio of polynomials), simplifying by factoring and canceling common terms may reveal the behavior near x = 2.
  
• Using Limit Laws: Special rules like the Dominated Limit Law or Squeeze Theorem allow indirect evaluation when direct substitution yields indeterminate forms.

For example, suppose g(x) = (4x² − 16)/(x − 2). Direct substitution gives 0/0 — an indeterminate form. But factoring the numerator:

Final Thoughts

g(x) = 4(x² − 4)/(x − 2) = 4(x + 2)(x − 2)/(x − 2)

Canceling (x − 2), for x ≠ 2, simplifies to 4(x + 2). Then:

limₓ→2 g(x) = limₓ→2 4(x + 2) = 4(4) = 16

Wait — here, limit is 16, not 4. But if instead g(x) = (4x − 8)/(x − 2), simplifying gives 4, so limₓ→2 g(x) = 4.

This illustrates why evaluating limits requires simplifying functions to expose asymptotic behavior near x = 2.