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Elon Musk has a plan for satellite internet that involves LEO and VLEO satellites. The question I have is how long would a satellite at these altitudes actually be within range before having to hand off to another? And how many satellites would it take at these altitudes to provide 24x7 coverage of a specific area?

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For how long is a satellite orbiting at 300 miles going to be visible from horizon to horizon?

Answer: 16.6 minutes

The formula for angular horizon distance is https://sites.math.washington.edu/~conroy/m120-general/horizon.pdf

$A = cos^{-1} \frac{R}{(R+h)}$ where:

A = angle

R = radius of Earth: 3,958 mi

h = altitude: 300 mi

A = 23°

Double that for horizon-to-horizon, and you get 16.6 minutes out of a 130 minute orbit.

The question I have is how long would a satellite at these altitudes actually be within range before having to hand off to another?

A lot less.

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    $\begingroup$ this answer is the maximum time between rise and set, which occurs only when the satellite passes directly overhead. the visible interval shrinks as the orbit track moves sideways, out of the plane of this picture. "actually being in range" is not necessarily less at all, but it needs more information to determine what the OP means by "in range". $\endgroup$
    – Ryan C
    Jan 25 at 22:39

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