Limited by human weakness refers primarily to human sinfulness and mortality.
God appointed his Son, Jesus, with an oath (see Psalm 110:4, which is important for understanding Hebrews 7:11–28).
In Hebrews, the term perfect means complete or mature (see 2:10; 5:9; 7:11; 9:9–11); the Son, through his sacrificial death and resurrection, has become completely qualified to serve as our eternal High Priest, and his priesthood lasts forever.