There aren't many direct applications of limits, but limits are used in other mathemathical branches such as infinitesimal calculus (derivations and integrals are defined as special limits of functions see e.g. Bartle, Sherbert: Introduction to Real Analysis). Calculating infinite sums (like Taylor or Fourier series), calculating limit of convergent matrix (which can be important for random process such as Markov chain), asymptotic analysis (which is important in design and analysis of algorithms see e.g. Knuth: The Art of Computer Programming vol. 1), finding convergece rate (important for design and analysis of numerical algorithms e.g. Newton method has superlinear convergence rate while Brent's has linear convergence rate which means that Newton's method is faster.), convergence of random variables (very important in probability theory and statistics adn as such in economics see e.g. Mittelhammer: Mathematical Statistics for Economics and Business).
Hope this helps.