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The end is near for car manufacturing firm, warns local council.
She is in the final stages of the disease, reveals her husband.

How would you describe the underlined parts? They are from newspapers. Are they phrases or sentence fragments: are they wrong in formal english.

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That is headline style. Actually, that type of phrasing screams headline style. It would sound like a headline no matter what the setting, so it is not suitable for ordinary writing, be it formal or informal.

Headline style eliminates unnecessary words. These do not do that so much, but they do change the word order to put the salient point first. The grammar is unusual but sound. Take the first one. It is equivalent to "The local council warns (that) the end is near for car manufacturing." Full-blown headline style would be more like "End near for car manufacturing, local council warns."