Slew Rate: The Speed the Output Cannot Keep Up With
Raise the frequency and the output loses the sine
The faint curve is the input sine wave and the bold gold curve is the output. Raising the input frequency makes the slope the sine demands steeper as well. At low speed the output follows the sine exactly, but at high speed the output can rise and fall only at a fixed maximum rate, so the sine distorts into a triangle wave. The multiple on the indicator is that demanded slope expressed as a multiple of the slew rate.
The output cannot change faster than a fixed speed
Inside an op-amp sits a compensation capacitor that prevents oscillation, and for the output voltage to change, this capacitor must be charged or discharged. But the internal current that does this job has an upper limit. The rate at which a capacitor's voltage changes is the current divided by the capacitance, so the maximum rate at which the output can change per unit time is fixed. This is the slew rate, usually written in volts per microsecond. As a formula, SR = Imax / C.
Too fast and the sine becomes a triangle
A sine wave is steepest at the instant it passes through 0, and that slope is proportional to the product of amplitude and frequency. When a large amplitude or a high frequency pushes this slope beyond the slew rate, the output cannot follow the sine's curve and only rises and falls along straight ramps of ±slew rate. So the output sine distorts into a triangle wave. This effect appears only with large signals. At the same frequency, a small amplitude demands a small slope and is followed without trouble, so the cause differs from the small-signal GBW limit. That is also why data sheets call the highest frequency at which a sine can be output at full amplitude without distortion the full-power bandwidth.
And the voltage limit: rail clipping
Large signals face a wall of height as well as a wall of speed. However hard it is pushed, the output cannot go beyond the supply rails. If you grow the amplitude until the peaks reach the positive or negative supply, everything past that is clipped flat; this is the output voltage limit, or rail clipping. Many op-amps already clip a little inside the rails, and there are separate rail-to-rail output devices built to swing close to the rails. In short, large signals in an op-amp meet two walls: the slew rate, which limits the rate of change per unit time, and the supply rails, which limit the height of the amplitude. Add the small-signal GBW, and the boundaries of what a real op-amp can do are all drawn.
Back to the first screen
At low frequency the output followed the input sine flawlessly. As you raised the frequency, the slope the sine demanded grew steeper, and from a certain point the output could no longer keep up, rising and falling only along straight ramps until it became a triangle wave. The boundary was whether the sine's maximum slope exceeded the slew rate, and beyond it even the output amplitude shrank. This is a wall of speed that appears only with large-amplitude signals, a different limit from the small-signal GBW. Add the wall of height, where the output cannot cross the supply rails, and the real op-amp hidden behind the ideal of infinite gain comes into view.
What comes next
You have now seen both the ideal and the limits of the op-amp. The last op-amp unit combines this device with resistors and capacitors to build useful circuits. Combine the way a capacitor's impedance changes with frequency with the op-amp's large gain, and you get an active filter that passes only the band of frequencies you want and suppresses the rest. It is a tool for picking out only the part of a signal you need.