One topic that often comes up in Death Note discussions is whether Light really cared about his family. I personally think Light loved his family, and here's my case for it!
While the manga focuses a little more on Light's relationship with his family than the anime, I think there's evidence for Light's love for them in both, so I'm going to hop haphazardly between the anime and the manga in my citations.
Early in the manga, Light and his sister Sayu seem to be pretty close. In chapter three, we see that Sayu is in the habit of coming into his room without knocking, that she admires him, and that she's comfortable enough with him to tease him about reading dirty magazines. Light helps her with her homework, and it seems to be a regular thing; their dad asks, 'Was Light helping you do your homework again, Sayu?' at the dinner table.
When Light learns that his family will see Ryuk if they touch the Death Note, he's afraid that this means he might end up having to kill them to protect his secret, so he builds a contraption to destroy the Death Note if it's ever discovered. This is a pretty huge thing for Light to do! He loves his supernatural murder method, he's built his entire vision of the future for both himself and the world around it, but he'd rather destroy it than have to harm his family.
Much later, Sayu is abducted and ransomed for the Death Note. Some people see that Light contemplates killing Sayu here and assume this means he would be fine with killing Sayu. But, if he didn't care about her, he'd have killed her straight away, just as he killed the director. In the anime, when we first see Light thinking that he might have to kill Sayu if this goes wrong, he's retreated into the bathroom to stare at himself in the mirror and he is clearly freaking out. He doesn't want to kill her! She's his sister, and he loves her!
When it comes down to it and Light has his pen over the paper, he hesitates in a way we've never seen him hesitate to use the Death Note before, and ultimately he lets her live. From a practical viewpoint, killing Sayu would have protected the notebook, and it would have reduced any remaining suspicion of Light; why would Kira kill his own family? But he can't bring himself to do it.
Light tells himself he shouldn't kill Sayu because not many people know about the abduction, but I don't think that's the only reason. I think Light is just desperately looking for reasons to convince himself that letting Sayu live is the right move, because he doesn't want her to die. Light lies to himself just as much as he does to anyone else, and the internal monologue we hear from him is often just what he's telling himself, rather than his actual thoughts and feelings.
Similarly, although Light occasionally thinks about killing the Kira investigation task force, he never actually follows through with it. Why? Because his dad is on the team, and he can't kill the rest of the task force and spare his dad; that would be an incredibly suspicious move. He has to kill all of them or leave them alone, so he leaves them alone.
Light's reluctance to harm the task force actually helps L deduce that Kira is related to someone on the team. In chapter 22 of the manga, L points out that, in addition to Kira having access to task force information, 'even though Kira murdered the FBI agents, he has not killed a single Japanese investigator'. Light can tell himself it's just not the right time, but L, as an outside observer, looks at Kira's actions and sees emotion there.
Light wants to think of himself as a person who's willing to do whatever it takes, so he tells himself he can kill the task force; when Sayu is abducted, he tells himself he can kill her. But, in the end, he can't bring himself to do it. It's very rare for Light to see a way a death could benefit his mission, to have the means to enact that death, and to let the moment pass by, but his family is the one area of his life where emotion consistently beats practicality for him.
If it really came down to it – if Light couldn't see any other way out – would he kill one of his family members? I think he might, but I think it would hurt him. And I think it's very telling that, in canon, when killing a family member could advance his goals, he always finds a reason not to.
The question of whether Light loves his family creates a lot of debate in the fandom. I think people often view the question as 'did Light care about his family or would he have been willing to kill them?' and end up arguing for one side over the other, but I personally believe that the answer is 'both'.