Look at the formal definition of Big O notation to see a limit defined within the notation there. While Big O is specifically about worst case complexity there are other notations for average case or best case.
Usually the idea is to get an upper bound of how bad does the algorithm run. Is the complexity logarithmic, linear, quadratic, or exponential? There are various examples one can give for some of those as a binary search can have logarithmic complexity as one is reducing half the options with each search assuming the initial set of data is sorted. Quadratic would be various poor sorting algorithms like Bubble Sort.
An alternative view here is to note that "the size of the input increases without bound" is generally a way of describing a limit as a variable tends to infinity. Limits to Infinity would be an example discussing f(x) = 1/x where as x grows without bound, the function value will approach zero. The key is to imagine the running time of the algorithm as a function depending on the size of the input and as one changes that input, how quickly is the function growing. Is it a straight line or a curve? How steep is the slope at any point? That is kind of the point behind the idea here.
