Thanks for using the forum. I am happy to address your questions, and my answer will be based on the information you provided here. Please make sure you recognize that this forum is for educational purposes only, and it does not substitute for a formal office visit with your doctor.
Without the ability to examine you and obtain a history, I can not tell you what the exact cause of your symptoms is. However I will try to provide you with some useful information.
I am not sure what you mean by dizziness, but if you mean vertigo, and you have vertigo associated with left sided numbness, there are many potential causes. The most concerning would be a stroke: the symptoms often start suddenly, and if a stroke occurs in a certain area of the brain called the brainstem, could theoretically involve the face, arm and leg (though the face symptoms would be on the opposite side of the arm/leg symptoms).
If by dizziness you mean vertigo, and if this is the most prominent symptom, with ringing in the ears, an inner ear problem could be a potential cause, though this would not explain your left arm and leg symptoms. Inner ear causes of vertigo most commonly include benign positional vertigo (BPPV), which is due to small particle in the inner ear that moves out of place, and can be repositioned with simple head maneuvers. The symptoms often include vertigo that occurs with turning of the head, often while turning over in bed. Another cause, if your symptoms are associated with tinnitus (ear ringing) and hearing loss is called Meniere’s disease and can be treated with medications and sometimes surgery. And so on, several other causes from inner ear problems exist.
If by dizziness you mean light-headedness, causes could include low blood pressure such as due to dehydration or autonomic dysfunction, cardiac problems, and several other non-neurologic causes. Anemia can cause light-headedness as well.
There are several other potential causes to your symptoms, and evaluation by your primary doctor (family physician/internist), with imaging of the brain is recommended.
Thank you for this opportunity to answer your questions, I hope you find the information I have provided useful, good luck.