Question:matzinu besipur shel shunamit, veelisha, ugejasi she lomdim min apasuk(o sheshunamit lomedet mimaase elisha) she elisha haya hish kadosh abal gejasi lo lefi shekeshedaja et shunamit naga bedadea abal im korim kol asipur shalem bemelajim mashma shegam ken gejasi haya ish kadosh bearehaya hu shebeshaa sheshunamit bikesh shetijaye et bena elisha shalaj et gejasi beim hu lo hish kadosh begadol ¿lama shalaj?ela mashama sheu haya gam ken kadosh beaguemara omeret sheu lo kol kaj?
[Translation: We find in the incident of the Shunamis woman and Elisha, that the verse teaches that Elisha was a holy person, but his attendant Gechazi was not, because he pushed the woman by her chest.
However, when we read the whole incident in Melachim, it seems as though Gechazi was also a holy man, for it was he whom Elisha sent to revive the child. If he was not a great person, why did Elisha send him? It must be that he, too, was Kadosh, and if so, why does the Gemara say that he wasn't?]
You are correct, as we find in a number of places in Sefer Melachim and in the Midrashim, which say that Gechazi was Elisha's chief disciple and pre-eminant attendant. However, the answer to your question can be found in the Yerushalmi (Sanhedrin 10:2) and in Pirkei d'Rebbi Eliezer (ch. 33), which state, "Gechazi was an Adam Gibor ba'Torah, but he had three bad character traits... and [third,] he was morally corrupt (Parutz b'Ervah), as the verse says, 'va'Yigosh Gechazi l'Hadfah...'."
It must be that his rebbi, Elisha, did not recognize these three bad traits of Gechazi. When Gechazi acted inappropriately with the Shunamis woman, either Elisha did not see this (for certainly Gechazi would not have been so brazen to do it in front of his rebbi), or he thought that it was done inadvertently, that the woman slipped or shook. Therefore, Elisha sent Gechazi to revive her son.
However, the end of the incident revealed what happened at the beginning: Gechazi did not fulfill his Shelichus and was unable to revive the child, as Rashi explains there (Melachim II 4:29).
Kol tuv,
Mordecai