These are classic examples of the "dangling participial phrase". The participle must modify the subject of the sentence. In sentence #1, who is "considering his abilities"?
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In the first sentence 'Considering his abilities', there is no subject of reference. Here the action 'Considering' has is left without a doer. We do not know who is 'Considering his abilities'. Hence,it is called impersonal absolute.
Rahul2689Please help me out with impersonal absolute participles wherein the participle is left without a proper 'subject of reference' ...
Impersonal absolute participle (clause) is a complicated name for a construction that appears to be a faulty participle clause