Limits and Continuity Practice

The relevant questions can be found here.

1. One should use a one-sided limit, more specifically a right-hand limit because \(sqrt(x)\) is only defined for \(x \geq 0\).
   

   \(\lim_{x \rightarrow 0^+} \sqrt{x} = 0\)
   

2. One should use a one-sided limit because \(\frac{1}{x+1}\) has an infinite discontinuity with asymptotes at \(x = -1\). Either side will work.
   \(\lim_{x \rightarrow 0^+} \frac{1}{x+1} = \infty\)
   \(\lim_{x \rightarrow 0^-} \frac{1}{x+1} = - \infty\)
   

3. One should use a two-sided limit because \(\frac{1}{(x-1)^4}\) goes to infinity as \(x\) tends toward one from either side due to its even asymptote.
   

   \(\lim_{x \rightarrow 1} \frac{1}{(x-1)^4} = \infty\)
   

4. One should use a two-sided limit because \(|sin(x)|\) is continuous.
   \(\lim_{x \rightarrow 1} |sin(x)| = 0\)
   
5. One should use a one-sided limit as the function \(\frac{|x|}{x}\) has a jump discontinuity at \(x=0\) and the left and right hand limits are not equal. Either side will work.
   \(\lim_{x \rightarrow 0^+} \frac{|x|}{x} = 1\)
   \(\lim_{x \rightarrow 0^-} \frac{|x|}{x} = -1\)