On one of my past calculus tests, I struggled with a multi-part question that asked to "Evaluate the limits that exist, or prove that they don't", as I would often be plagued by doubt wondering if each limit exists, and wound up wasting time changing my answers.
Now with my final upcoming later this week, the last thing I want to do is spend any time second-guessing myself as to whether or not the limit exists. What sort of techniques can I use to intuitively determine the existence of a limit of a function before delving into evaluation or a proof of non-existence?
I'm in a first year single variable Calculus course; here are a few limits.
$\lim_{x \to 0} \frac{\tan(3x^2) + \sin^2(5x)}{x^2}$
$\lim_{x \to 3^+} \frac{\sqrt{x - 3}}{|x - 3|}$
$\lim_{x \to 1} \frac{x^2 - \sqrt{x}}{x - 1}$