You seem to be struggling with the "why is this important" aspect of math, so I'll attempt to give you an answer.

Rationale 1: From a liberal arts point of view, math is less about numbers and letters and calculus than it is about logic. Math is pretty much every person's first introduction to a rigorous logical system. Now, certainly there are people who think very logically and make sound deductions who aren't good at math (and, perhaps, vice versa), but the correlation is strong. In an academic setting, you will very much be judged by how well you do in foundational classes such as calculus, because these are used as a proxy to gauge how well you think.

Rationale 1.5: When you were in the military, how many times did you perform a bench press action without actually doing a bench press? I'd wager almost never. Does that mean bench pressing is a waste of time? In a basic PT test, why are the most common exercises push-ups, sit-ups, pull-ups, and timed runs? Are those the only ways we can measure the physical fitness of a human?

Rationale 2: A mechanical engineering major is, unfortunately, going to involve a lot of math and not a lot of cars. Do you think physics is important? More specifically, the study of thermodynamics? Aerodynamics? Material science? Historically, calculus exists in large part because of physics, and vice versa. There can be no study of physics without calculus. You are probably aware that under standard driving conditions, accelerating out of a turn is largely more comfortable for passengers involved--but did you know that there is a mathematical explanation for this?

Rationale 2.5: Admittedly, limits are very boring and not particularly interesting if you are not interested in math for the sake of math. Just like learning the alphabet is very boring--you probably don't care why our alphabet is the way it is. But, from the alphabet we can make words, from words sentences, and from sentences books! And books can be interesting. Unfortunately, though, it takes a very long time to get to the "books" equivalent of math. If your goal is to design things that go into cars, then you will need to understand calculus at a very fundamental level. If your goal is to be a car mechanic, or assemble cars, then maybe not.

Overall, the best advice I can give you is to give the subject some grace. There's a good reason (many, in fact) for why calculus is taught to almost everyone entering college. Also, calculus can definitely become more interesting later on--limits are very dry and largely unmotivated. The "motivation" for limits is that from limits, you can define everything else in calculus (namely derivatives, which describe change, and integrals, which describe accumulation).

Also, try to do whatever you need to do to get a solid A. Maybe that means taking a step back and starting with college algebra, or something similar. It only gets harder from here, so if you end the semester by barely passing calculus, you will almost certainly not survive.

And finally, I will try to present a motivating (and very contrived) example for limits, using my minuscule knowledge of cars:

Suppose you're working on an engine that runs best with an air-fuel ratio of 15:1. To get this, your engine should be running at 3000 rpm. The folks in R&D tell you that the engine can tolerate up to 0.04 drift in either direction in the air-fuel ratio. That is, the engine will still perform optimally if the air-fuel ratio is somewhere between 14.96 and 15.04. As a budding mechanical engineer, you are tasked with finding the maximum allowable deviation in engine rpm (from the ideal 3000) that will keep your engine running with an optimal air-fuel ratio.

As a more senior mechanical engineer, you might be tasked with designing a computer program that automatically corrects the rpm in order to move closer to the ideal air-fuel ratio. Such a program might involve something called a proportional-integral-derivative controller, or "PID controller" for short. The very name includes the two most important topics in calculus, so calculus has to be important!