When are graphs deceiving? What are some examples of functions or quantities relating to functions (e.g., limits) $f:A \to B$ where $A$, $B \subseteq \mathbb{R}$ that require by-hand, "analytical" methods for analysis which are seemingly contradicted by a graph generated by software?
For instance, I recall that a Pre-calculus text stated that $\lim\limits_{x \to 0}\dfrac{1-\cos x^{6}}{x^{12}} = \dfrac{1}{2}$ (which, if I recall correctly, is proven using Taylor series) but the graph itself seems to suggest that it perhaps doesn't exist, due to the oscillations occurring around 0. [Graphs were generated via WolframAlpha.]



Almost any example of catastrophic cancellation plus enough zoom will do the trick. Two cases:
A simplification of your example (simpler function, bigger zoom):
$$\frac{(1-\cos(x^2))}{x^4},\qquad x\in[-0.001,0.001]$$

A rational function with a removable discontinuity:
$$\frac{x^{50}-1}{x-1},\qquad x\in[0.999999999,1.000000001]$$

