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Sorry if my grammar is a bit off. But I have few questions about space and its expansion.

If space expands faster and faster all the time, will there be a time when we cannot see further than "hubble sphere"?

Am I correct by thinking that we cannot ever reach the edge of the current hubble sphere due to not being able to go faster than a speed of light and because the edge will be expanded further away?

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No, we will always be able to see farther than the Hubble sphere (in theory).

This spacetime diagram — taken from Pulsar's rendering of Davis & Lineweaver (2003)'s Figure 1, in this excellent answer — can help visualize it:

Spacetime

Coordinates

In this figure, time increases upward, we're the vertical line in the middle, Big Bang is the bottom line, and our current time is the black horizontal line. The $x$ axis shows distance from us in comoving coordinates, i.e. the coordinates that expand with the Universe, and in which galaxies lie approximately still. By definition, today comoving coordinates coincide with physical coordinates (i.e. the "real" distance you would measure if you stopped time and laid out rulers), but for instance 8 billion years (Gyr) ago (indicated on the left $y$ axis), when the scale factor $a$ was $0.5$ (indicated on the right $y$ axis), the distance between two given galaxies in physical coordinates would be half that in comoving coordinates.

Particle horizon

The farthest distance to which we can see is called the particle horizon, and is drawn in blue. It will always increase, since light from farther and farther away eventually will reach us (although light from these distances will be increasingly more redshifted and eventually will be unobservable in practise).

Hubble sphere

In contrast, the Hubble sphere — i.e. the distance at which space expands at the speed of light — doesn't increase much more than today in physical coordinates. In comoving coordinates, the Hubble sphere reached a maximum roughly 5 billion years ago, and is now decreasing (the innermost green solid line labeled $v_\mathrm{rec}=c$). In other words, galaxies closer and closer to us are receding at $v=c$, but not until they are farther away from us physically.

Can we reach the Hubble sphere?

As for your second question, the answer depends a bit on what you mean: By definition, the Hubble sphere is a certain distance from you, so taking your question literally, it's impossible to go to the Hubble sphere, since it will always be some distance from you. But I assume that you mean "Is it possible for us the send a space probe from Earth, out to Earth's Hubble sphere". The answer to this is "Yes". Although special relativity prohibits the space probe from going faster than $c$ locally, as seen from Earth the expansion of space "helps" the probe to go faster and faster, eventually surpassing $c$ (just like it helps galaxies going faster than $c$).

In the figure, the orange dashed line is our future light cone; events outside this region can never be affected by us. You see it incercepts the Hubble sphere approximately at $t=25\,\mathrm{Gyr}$, or $a=2$. This means that, if a space probe departs today and travels at almost the speed of light, it will reach our Hubble sphere in 10 billion years or so, at which time the Universe has doubled its size (in all three directions). A space probe traveling at more realistic velocities will reach the Hubble sphere later; the exact time can be seen from the figure using a wordline with a slope that is steeper than the 45º that light rays have.

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