Limits

Normal or "easy" limits are rather simple, as examples like \(\lim_{x\rightarrow4} \frac{x+3}{x^2 + 1}\) just need some plugging in.
Derivatives are usually harder as \(\lim_{x\rightarrow0} \frac{f(x_0+\Delta x) - f(x_0)}{x-x_0}\) always evaluates to \(\frac{0}{0}\), and needs some cancellation.

DEFINITION Right hand limit or \(lim_{x\rightarrow x_0^+} f(x)\) indicates that \(x\) is greater than \(x_0\) (or that \(x\) begins on the right side of the number line).

DEFINITION Left hand limit or \(lim_{x\rightarrow x_0^-} f(x)\) indicates that \(x\) is less than \(x_0\) (or that \(x\) begins on the left side of the number line).
These notations will make dealing with limits of these functions more convienient.

EXAMPLE
Take the following example of a conditional function:

if \(x > 0\), \(f(x) = x+1\)

if \(x < 0\), \(f(x) = -x+2\)

\(lim_{x\rightarrow x_0^+} f(x) = lim_{x\rightarrow x_0} x+1 = 1\)

\(lim_{x\rightarrow x_0^-} f(x) = lim_{x\rightarrow x_0} -x+2 = 2\)

We did not need a \(x=0\) value to compute these limits!

A checklist for what to do before dealing with nested limits.

EXAMPLE: \(\sin{\sqrt{x}}\)

• [ ] Check domain + range of inner function (in this case \([0, \infty)\), \([0, \infty)\)).
  
• [ ] Check domain + range of outer function as well as what it takes in. (takes in \([0, \infty)\), range is \([-1, 1]\))
  
• [ ] Restrict domain based on requirements of inner + outer functions
  

EXAMPLE: \(\ln{\sin{x}}\)

• [X] Domain of \(\sin{x}\) is $(-∞, ∞), range is \([-1, 1]\).
  
• [X] Domain of \(\ln{x}\) is \([0, \infty)\), range is \((-\infty, \infty)\).
  
• [X] As \(\ln{x}\) takes only positive values, the restricted domain for the composite function is \([0, \pi]\), \([2\pi, 3\pi]\), etc. The range of the composite function would be \((-\infty, 0]\).
  

Adjacent to this: Continuity

Building upon this: Calculating Derivatives

Further reference can be found at Limits and Continuity Practice.