Heartrate

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I'm a 45 year old female. Whenever I do any standing cardio (treadmill, elliptical), my heart rate immediately jumps into the 180s. This happens within five minutes of working out and I don't feel like I'm pushing myself at all or tired. At times I'll feel really awful when this happens. Not quite nauseous, but just yucky. I have to go very slowly just to keep it in the 160s and even then it creeps up. I mostly use the recumbent bike as a result as I feel I have better control that way and keep my heart rate in the 140-50s. Again, though, I have to go very slowly and am not really pushing myself to keep it at those rates. My resting heart rate is typically about 110. The lowest I've ever seen it was 87.

I've just been diagnosed with very severe anemia. Are these typical rates for someone with anemia? As my iron levels go up can I expect my heart rate to lower? Is it okay to just ignore the heart rate and push myself?