I often recommend to my students that they follow their noses when solving mathematical problems. The role of intuition in creative mathematical work is difficult to explain. Still its value should never be underestimated and its development may be fostered by experience, including even unsuccessful trips down dead ends and meandering forays into thickets of apparently untameable complexity. In this article I invite you to corne with me and explore a simple calculus problem using any tools and ideas that seem to offer some promise of moving us along in the right direction. The scenic route embraces Taylor's formula, geometric series expansions, relations in Pascal's triangle, and matrix diagonalisation. Don't forget to bring along a pencil and some scrap paper… and a computer algebra system too, if you like.