Split into right and left limits.

Right limit ($x\to0^+$): $[x]=0$, so $\sin(x[x])=0$, $|x|=x$.

$\dfrac{\tan(\pi\sin^2x)+x^2}{x^2}$. Using $\tan(\pi\sin^2x)\approx\pi\sin^2x\approx\pi x^2$:

$=\pi+1$

Left limit ($x\to0^-$): $[x]=-1$, $\sin(x[x])=\sin(-x)=-\sin x$, $|x|=-x$.

$=\dfrac{\tan(\pi\sin^2x)+(-x-(-\sin x))^2}{x^2}=\dfrac{\pi x^2+(\sin x-x)^2}{x^2}\to\pi+0=\pi$

Left $\neq$ Right, so the limit does not exist.

The textbook discusses noise in the Shannon-Weaver model context and identifies three categories:

1. Physical (Mechanical/Engineering) Noise: Unexplained variation or random error in the communication channel. Examples: a loud motorbike during conversation, smudges on a printed page, “snow” on a TV set, mist on a car windscreen, a crackling microphone. Shannon was primarily concerned with this type.
2. Semantic Noise: Arises from differences in meaning — when sender and receiver do not share the same knowledge, cultural background, experience, attitudes, beliefs, or linguistic skills. This is “the very essence of the study of human communication.”
3. Psychological Noise: The textbook mentions “psychological interference with encoding and decoding” — arising from emotional states, biases, prejudices, or mental conditions that distort how messages are encoded or decoded.

Channel overload — when the channel capacity is exceeded — is also mentioned as a source of distortion distinct from noise per se.