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We report a new design of carbon nanotube (CNT) resonator, whose resonance frequency can be tuned not only transversally by a gate voltage, but also by the axial strain applied through directly pulling the CNT. The resonators are fabricated from individual suspended single-walled CNT (SWCNT) in situ inside a scanning electron microscope. The resonance frequency of a SWCNT resonator can be tuned by more than 20 times with an increase of quality factor when the axial strain of the SWCNT is only increased from nearly zero to 2% at room temperature. The transversal gate-tuning ability is found to be weaker than the axial-tuning ability and decrease with increasing the axial strain. The gate voltage can hardly tune the resonance frequency when the initial axial strain is larger than 0.35% and the CNT acts like a tied string. The relationship among resonance frequency, gate voltage, and initial axial strain of the CNT obtained presently will allow for the designs of CNT resonators with high frequency and large tuning range. The present resonator also shows ultrahigh sensitivity in displacement and force detection, with a resolution being better than 2.4 pm and 0.55 pN, respectively.