How does a gravitational redshift differ from a doppler shift?
A “gravitational redshift” is the frequency (speed) change of a photon (light) in a gravitational field.
When light and sound are components of a moving object (planes, planets, trains, etc.), they both exhibit doppler shifts: the incoming perspective is compressed, or "blue shift," and the outgoing (receding) perspective is expanded, or "red shift." A "Doppler shift" is a more general term for the phenomenon of a wavelength shift caused by external forces.
Other electromagnetic wavelengths will show similar shifts, but they won't necessarily become "redder." The term "red shift" refers to the movement of light frequencies to the longer "red" wavelengths.
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A gravitational redshift is caused by differences in gravitational potential between the source and the observer, leading to a shift in the wavelength of light. A Doppler shift, on the other hand, is caused by relative motion between the source of light and the observer, resulting in a shift in the frequency of the light.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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