Find the limit as 𝑥 tends to negative one of 30.
This is a constant function 30, the function that returns the output 30 no matter what input you give it. And we have to find the limit as 𝑥 tends to negative one of this function. We have a rule for this limit. The limit as 𝑥 tends to 𝑐 of the constant function 𝐾 is just 𝐾. We apply this to the limit we want to find, where 𝑐 is negative one and 𝐾 is 30. The limit and hence our answer is 30.
Let’s have a look at the graph of the function to see why this value makes sense. The graph of our function is a straight line parallel to the 𝑥-axis with equation that 𝑦 equals 30. We wanted to find the limit as 𝑥 tends to negative one of this function. And so we draw in the line 𝑥 equals negative one. As 𝑥 gets closer and closer to negative one, the value of the function remains the same at 30. This is a constant function after all.
The same is true as 𝑥 approaches negative one from above, that is, with values of 𝑥 greater than negative one. We sometimes talk about how 𝑓 of 𝑥 gets closer and closer or tends to or approaches a certain value, as 𝑥 approaches negative one in this case. But if 𝑓 of 𝑥 stays the same at 30, then we’re not really approaching 30 or getting closer and closer to 30. We’re just staying the same.
The language we use is supposed to help us understand what a limit is intuitively. But in this case, it looks like the language is getting in the way of the unambiguous and objective truth that we aim for in mathematics. Just going by the language, someone could make an argument that the value of the limit should not be 30 as 𝑓 of 𝑥 is not getting closer and closer to 30. It’s just staying the same. That’s why we need unambiguous objective rules like the limit as 𝑥 tends to 𝑐 of the constant function 𝐾 is 𝐾.
This isn’t to say that we should throw away our intuition about limits. Rather, we should use our intuition about limits to write down and understand the laws that limits should follow. This is like how intuition about morality and justice allows legislators to decide what the laws of a country should be. And our intuition about morality and justice allows us to understand why those laws are there rather than just seeing them as arbitrary rules to follow.