Indeterminate forms involving fractions

If you look at $$\lim_{t\to{0}}\left(\frac{1}{t} - \frac{1}{t^2+t}\right)\quad\text{  or  }\quad\displaystyle\lim_{x\to{-4}}\frac{\frac{1}{4} + \frac{1}{x}}{4+x}$$ you will get the forms $\infty-\infty$ and $\tfrac00$. 

There are ways to simplify these fractions.  For example, we can change the form of the function that is a sum or difference of fractions by finding a common denominator. For instance, $$\frac{1}{4} + \frac{1}{x}= \frac{x}{4x} + \frac{4}{4x} = \frac{x+4}{4x},$$ and $$\frac{1}{t} - \frac{1}{t^2+t} = \frac{t+1}{t^2+t}-\frac{1}{t^2+t} = \frac{t}{t^2+t} = \frac{1}{t+1} \left(\text{ since } t\ne 0\right).$$ These are examples of how to do more work when you get indeterminate forms $\infty-\infty$ and $\tfrac00$ involving fractions.

The following video shows the details of evaluating these limits.