{'fraction_non_alphanumeric': 0.015113350125944584, 'fraction_numerical': 0.0, 'mean_word_length': 3.975, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The Limit of a Real Function at a Point

Summary. We define the proper and the improper limit of a real function at a point. The main properties of the operations on the limit of a function are proved. The connection between the one-side limits and the limit of a function at a point are exposed. Equivalent Cauchy and Heine characterizations of the limit of a real function at a point are proved.