Why does the bronchodilator exaggerate the tachycardia?

Answer 1

Most bronchodilators, such as albuterol (aka salbutamol in Europe), are modifications of adrenaline (epinephrine). There are several different types of adrenaline receptors in the body and the medications bind better to the type that opens up the lungs than to the type that speeds up the heart. That, and the fact that you deliver more drug to the lungs than the heart by inhaling them, means that the effect of the medication is much greater in the lung (bronchodilation) than in the heart (fast heart rates - tachycardia).

This is not perfect. Some is absorbed into the blood, especially if the inhaler is used a lot, and so gets to the heart and makes it go fast.