Understanding $\lim_{x \to c} f(x)$, Pathologies, and $\frac{0}{0}$ Indeterminate Forms
$f(x) = \frac{x^2 - 1}{2}$
Touch / Drag Canvas
Insert Hole at $x=c$:
Position $x$:1.20
Left ($x < c$)Target c = 2Right ($x > c$)
Approach $x$:0.50
Microscope Zoom ($1\times - 100\times$):10x
Position $x$:1.50
Approach 2 from LeftHole at x = 2Approach 2 from Right
Readout & Evaluation
Left Limit $\lim_{x \to c^-} f(x)$:1.500
Right Limit $\lim_{x \to c^+} f(x)$:1.500
Overall Limit $\lim_{x \to c} f(x)$:1.500
Exact $f(c)$ at target $c$:1.500
Bridge to the Derivative
When evaluating the derivative $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$, direct substitution of $h = 0$ always yields the indeterminate form $\frac{0}{0}$! As shown by the dashed curves above, both the numerator $N(h)$ and denominator $D(h)$ collapse to zero simultaneously. The limit resolves this $\frac{0}{0}$ hole into a well-defined finite number: the exact instantaneous tangent slope.