Why does the (relativistic) mass of an object increase when its speed approaches that of light? I'm reading Nano: The Essentials by T. Pradeep and I came upon this statement in the section explaining the basics of scanning electron microscopy.

However, the equation breaks down when the electron velocity approaches the speed of light as mass increases. At such velocities, one needs to do relativistic correction to the mass so that it becomes[...]

We all know about the famous theory of relativity, but I couldn't quite grasp the "why" of its concepts yet. This might shed new light on what I already know about time slowing down for me if I move faster.
Why does the (relativistic) mass of an object increase when its speed approaches that of light?
 A: The mass (the true mass which physicists actually deal with when they calculate something concerning relativistic particles) does not change with velocity. The mass (the true mass!) is an intrinsic property of a body, and it does not depends on the observer's frame of reference. I strongly suggest to read this popular article by Lev Okun, where he calls the concept of relativistic mass a "pedagogical virus".
What actually changes at relativistic speeds is the dynamical law that relates momentum and energy depend with the velocity (which was already written). Let me put it this way: trying to ascribe the modification of the dynamical law to a changing mass is the same as trying to explain non-Euclidean geometry by redefining $\pi$!
Why this law changes is the correct question, and it is discussed in the answers here. 
A: Sometimes the same word "mass" is used with different meanings. There are two different quantities associated with the word "mass":


*

*A quantity that physicists usually call "mass", which is an intrinsic property of the object and does not depend on how fast it is moving. I'll use the symbol $m$ for this quantity.

*A synonym for the object's energy $E$, but expressed in mass-like units as $E/c^2$. This is sometimes called the object's "relativistic mass" and it does depend on how fast the object is moving (because the object's energy does).   I'll use the symbol $m_R$ for this quantity.
We are already familiar with the fact that the kinetic energy of an object is higher when the object is moving faster. "Relativistic mass" is just a synonym for the object's total energy, expressed in mass-like units. From this perspective, consider the question again:

Why does the (relativistic) mass of an object increase when its speed approaches that of light?

Answer: Because the object's energy increases. "Relativistic mass" is just a synonym for the object's energy, expressed in mass-like units. Why did people ever start using the name "relativistic mass" for the object's energy?  I don't know. In my experience, most physicists just call it energy.
Here are some equations to help clarify things:
The energy $E$, momentum $p$, speed $v$, and mass $m$ of an object are related to each other according to these equations:
$$
   E^2 - (pc)^2 = (mc^2)^2
\hskip2cm
  v = \frac{pc^2}{E}
$$
where $c$ is the speed of light. The $m$ in the first equation is what physicists usually mean when they use the word "mass". It is an intrinsic property of the object and does not depend on the object's speed.  The object's energy $E$ and momentum $p$ do depend on the speed, and they do so in such a way that the combination $E^2-(pc)^2$ does not depend on the speed. That's why this particular combination is interesting, and that's why the $m$ on the right-hand side of the equation deserves a special name: mass.
To relate this to the "relativistic mass" $m_R$ (which, again, is not used by the majority of physicists in my experience), re-arrange the second equation shown above to get
$$
   p = \frac{E}{c^2}v.
$$
If we  use $m_R$ as an abbreviation for $E/c^2$, then this becomes
$$
   p = m_R v,
$$
which looks superficially like the more familiar low-speed approximation $p=mv$. This resemblance is also misleading, though, because the energy $E$ (and therefore $m_R$) is a function of $v$. The momentum $p$ is not really proportional to the velocity $v$, except approximately when $v\ll c$. 
A: In special relativity the actual invariant is the magnitude of the covariant energy momentum 4-vector $(E_0/c_0, p_x,p_y,p_z)$, not the apparent mass itself.  See also the section on "momentum in 4 Dimensions", here.  The apparent mass in a moving frame is just a projection.
A: Keeping it simple (with a link): 
Special Relativity
"The relativistic increase of mass happens in a way that makes it impossible to accelerate an object to light speed: The faster the object already is, the more difficult any further acceleration becomes. The closer the object's speed is to light speed, the greater the increase in inertial mass; to reach light speed exactly would require an infinitely strong force acting on the body. This enforces special relativity's speed limit: No material object can be accelerated to light speed. 
The increase in inertial mass is part of a more general phenomenon, the relativistic equivalence of mass and energy: If one adds energy to a body, one automatically increases its mass; if one takes energy away from it, one decreases its mass. In the case of acceleration, the object in question gains kinetic energy ("movement energy"), and this increase in energy automatically means an increase in mass."
See http://www.einstein-online.info/elementary/specialRT/emc
This, to most, helps clear things up without adding complexity. You are, of course, welcome to delve deeper.
A: There is a point of view, that under the term "the mass" one must mean "the rest mass".
From that point of view there is obviously no dependence of the (rest) mass on the speed of an object. And, therefore, the mass of an object does not increase when its speed increases.
The correct (from that point of view) way to talk about the phenomenon is to say that with increase of the speed of an object you need more and more energy in order to make it move faster. 
Of course there is no fundamental controversy between this point of view and that of many books and articles. But the usage of the concept of "relativistic mass" makes things much more complicated, even if it was introduced in pursuit of simplicity. 
A: The complete relevant text in the book is

The de Broglie wave equation relates the velocity of the electron with its wavelength, $\lambda = h/mv$ ... However, the equation breaks down when the electron velocity approaches the speed of light as mass increases. ...

Actually, the de Broglie wavelength should be
$$ \lambda = \frac hp, $$
where $p$ is the momentum. While $p = mv$ in classical mechanics, in special relativity the actual relation is
$$ \mathbf p = \gamma m \mathbf v = \frac{m\mathbf v}{\sqrt{1-\frac{v^2}{c^2}}} $$
where $m$ is the rest mass. If we still need to make the equation $p = mv$ correct, we introduce the concept of "relativistic mass" $M = \gamma m$ which increases with $v$.
A: Fundamentally, mass and energy are the same thing. They are two "points of view" of the same reality.
From the "point of view" (inertial frame) of an electron, its mass does not increase, its speed is always zero.
From the "point of view" (inertial frame) of a stationary observer, the electron has a very high kinetic energy (some in the mass term and some in the speed term)
From the "point of view" (inertial frame) of a moving observer, the electron has a different kinetic energy (some in the mass term and some in the speed term)
And so on.
A: If you want to intuitively see why the mass increases, consider the following.


*

*Firstly, nothing can travel faster than the speed of light (this is the premise on which Special Relativity is based)

*Secondly, applying a force to an object will increase its kinetic energy (assuming the force acts in the same direction as the object's motion)
Since kinetic energy $K.E.$ = $m v^2/2$, if $v$ is limited to $c$, then as $v$ approaches $c$ the only way for $K.E.$ to increase is for $m$ to increase.
This isn't a fully mathematical answer, but may help you to intuit why the mass increases.
A: The reason why you are having this confusion is because you think that mass should not change. As many have said above, and I would reiterate, REST MASS is the property that does not change for any particle, ever. For eg, the rest mass of a photon is zero. So, basically, when einstein put forward the very famous equation, $E = M.C^2$, he meant very clearly that mass IS energy, and energy IS mass. They are just one and the same thing!. 
Now, tell me, if energy increases, would the mass not increase? And why not in daily life, the answer is because $ \delta M =  \frac{\delta E}{c^2}$...and so, if your energy changes by an amount comparable to $c^2$, only then would you be able to observe a change in mass.
Hope it helps...if any more doubts arise, please comment! 
A: Relativistic mass, by definition, is the quantity $m_\mathrm{rel}(v) := \gamma(v)\ m$, where $m$ is the intrinsic, or "rest", mass. Mathematically, it increases because the Lorentz factor $\gamma(v)$ increases with increasing speed $v$.
However, a more physical reason is that this "relativistic mass" is really just the total energy $E_\mathrm{tot}$, consisting of the combination of the rest energy and kinetic energy of a material body, interpreted in units of mass, by the mass-energy equivalence relation $E = mc^2$: $m_\mathrm{rel}(v) = \frac{E_\mathrm{tot}(v)}{c^2}$. It is thus seen to increase because objects moving at higher speeds have more kinetic energy, and thus also, total energy.
A: Hope the following helps, too:
For the "why" think about Einstein's photon from the sun being deviated by gravitation of the moon during some eclipse of the 20th century. The photon's direction was deviated by gravitational force. What about its velocity? Had it been accelerated by gravitational mass?
The answer is "of course not" - by definition there is no speed exceeding speed of light the photon brought along from the sun.
Thus, it comes to no surprise that the following is prone to be considered falsely claimed: If a photon is accelerated (by a gravitational field) a   shift of frequency occurs. There does exist red shift and - even -  violet shift of frequency, thus there does exist loss and gain of energy, shown by change of frequency, not velocity of the particle. This gain of energy (the cause  is acceleration, the consequence is energy)  may be named gain of mass as the change of energy the particle bears will  translate into Newtonian  momentum should the particle "hit any floor" (sun sail).
Velocity, in fact, is what the question is about. Why does it take that much to "accelerate" to speed of light. The question does not ask: Why is it so difficult to pump up the energy of a particle moving closer to c. The Why is about the speed, velocity. The name is "mass" because particles moving close to c will impart regular impulse, thus regular acceleration, velocity, when they interact with masses (hitting any floor).
Assuming that any particle close to c has the zero rest mass property of the photon, I think one way to answer about the "Why" is:
The closer an object moves at speed of light the more that particle is able to transform  any input of kinetic energy into non-kinetic energy ("frequency", "wave function"). This energy   may be called mass as that non-accelerated particle or object is able  to transform this non-kinetic energy into kinetic impulse (only "rest" mass of photons is zero) when "hitting".
The question "Why" thus asks about a mechanism that transforms kinetic impulse into energy (named mass) and energy to kinetic impulse which in my opinion is not known.
A: There are plenty of misinformation here.

"The mass of a body is not constant; it varies with changes in its energy."
[Einstein, A. The Meaning of Relativity, Princeton University Press, 1988]

See also Section 10, Dynamics of the Slowly Accelerated Electron, of the paper 'On the Electrodynamics of moving Bodies' [Einstein, A. Annalen der Physik, 17, 1905]. Also see Section 29, Ponderomotive forces. Dynamics of the electron, in the book 'Theory of Relativity' [Pauli, W. Dover Publications Inc., 1981, (first published in 1921 in German, first published in English in 1958)]
A: If I accept true mass as being constant then defining increases to such a given mass is impossible (see Newton) because I can sit and watch it forever at rest or in motion: a given mass is a given mass, and a given quantity does not change, regardless of speed or energy applied. The effect(s) then, either perception or theoretical, that creates the illusion that mass changes, is better described by a more accurate theory than Einstein put forth or modifying Einstein to a more accurate level. 
On another note, the arbitrary  Einsteinian speed of light limit for velocity is similar to the ceiling of the 4 minute mile. There is no proof that such a limit exists. Until we get past this arbitrary barrier and explain these phenomena in a more useful way we will never achieve deep space travel. 
To make such travel possible we must get beyond mass particles into non-mass particles that pass easily through the light barrier. They might or might not exist at this point. In the future these particles will be the building blocks and the carriers of information necessary to transfer life to habitable planets throughout the universe. Granted, your body will not beam up but your current conscionsness and DNA sequence might: A quick resurrection and you are you three galaxies away. 
Granted that I am not a physicist and I am an iconoclast when it comes to accepting dogma. To me, devolving honest perception into theoretical soup does not make you any more informed on the true nature of mass vis a vis energy and velocity. You simply have learned what someone else learned and thought it to be true, i.e. the appeal to authority that Einstein believed and said it therefore it is true. 
A: The mass of object changes when its speed approaches zero because according to Einstein postulates of theory of relativity all the laws are same in all inertial frames and speed of light remains constant in inertial frame in vacuum. All the concepts of relativity are based on these two postulates. As one can not add any speed in speed of light, the Lorentz transformation equations are derived and using these variation of mass with velocity relation. Almost every concept of Physics changes at a speed comparative to speed of light.
One can see the derivation here
