Vertical Asymptote

What happens to the image distance inside a camera as an object moves closer to the lens's focal length?

For a camera lens, the object distance x, image distance y, and focal length F are related by the thin-lens equation. Rewriting this equation in terms of y gives a rational function.

For any rational function, a vertical asymptote can occur where the denominator equals zero, provided that factor doesn't cancel with the numerator.

Suppose the lens has a focal length of 55 mm. The denominator becomes zero when x = 55, making the function undefined.

For this physical model, we consider object distances greater than the focal length, where a real image forms behind the lens. So, the relevant domain is x greater than 55 millimeters.

As x approaches 55 from values greater than 55, the denominator becomes an extremely small positive number, causing y to grow without bound. On the graph, the curve shoots upward sharply near x = 55, creating a vertical asymptote.

Physically, as the object approaches the focal length, the sharp image forms farther behind the lens. Since the sensor stays at a fixed position, the image becomes blurry.