Onesided limits require a good understanding of piecewise functions. If you are a little rusty or just need a quick reminder, you can find a complete discussion of piecewise functions on the piecewise functions precalculus page.
Onesided limits are finite limits where we evaluate the limit from each side of a point individually. The notation we use is
\(\displaystyle{ \lim_{x \to a^}{f(x)} }\) 
evaluate the limit on the left side of \(a\), i.e. values of \(x < a\) 
\(\displaystyle{ \lim_{x \to a^+}{f(x)} }\) 
evaluate the limit on the right side of \(a\), i.e. values of \(x > a\) 
The negative and positive sign that look like exponents on the finite value \(a\) indicate the side that we are looking at.
One of the reasons we need to look at limits on both sides of some number is when we are determining continuity. As you know from the continuity page, one of the requirements for continuity is that the limit at a point must exist. In order for a limit to exist, the limit from the left must be equal to the limit from the right, i.e. \(\displaystyle{\lim_{x \to a^}{f(x)} = \lim_{x \to a^+}{f(x)}}\).
Notice that we are NOT saying that the function value must be equal to the limit or even that the function need be defined at \(x=a\), only that the limit be equal on both sides of a.
Here is a great video to build your intuition of onesided limits. He uses an absolute value function to discuss the idea of onesided limits and limits that do not exist.
video by Dr Chris Tisdell 

Here is a good video showing a graph with several onesided limits.
video by Krista King Math 

You will see more onesided limits when you learn about continuity. For now, let's work these practice problems.
Practice
Unless otherwise instructed, evaluate the following limits. Give your answers in exact form.
Basic Problems
\(\displaystyle{\lim_{x\to1}{\frac{x1}{x1}}}\)
Problem Statement 

Evaluate the limit, giving your answer in exact form. \(\displaystyle{\lim_{x\to1}{\frac{x1}{x1}}}\)
Solution 

video by PatrickJMT 

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\(\displaystyle{\lim_{x\to0^}{\left(\frac{1}{x}\frac{1}{x}\right)}}\)
Problem Statement 

Evaluate the limit, giving your answer in exact form. \(\displaystyle{\lim_{x\to0^}{\left(\frac{1}{x}\frac{1}{x}\right)}}\)
Solution 

video by PatrickJMT 

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\(\displaystyle{\lim_{x\to5^+}{\frac{6}{x5}}}\)
Problem Statement 

Evaluate the limit, giving your answer in exact form. \(\displaystyle{\lim_{x\to5^+}{\frac{6}{x5}}}\)
Solution 

video by PatrickJMT 

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\(\displaystyle{\lim_{x\to0}{\frac{x1}{x^2(x+2)}}}\)
Problem Statement 

Evaluate the limit, giving your answer in exact form. \(\displaystyle{\lim_{x\to0}{\frac{x1}{x^2(x+2)}}}\)
Solution 

video by PatrickJMT 

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\(\displaystyle{\lim_{x\to4^}{x+4}}\)
Problem Statement 

Evaluate the following limit, giving your answer in exact form. \(\displaystyle{\lim_{x\to4^}{x+4}}\)
Solution 

video by PatrickJMT 

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\(\displaystyle{\lim_{x\to5^+}{\sqrt{x^225}}}\)
Problem Statement 

Evaluate the limit, giving your answer in exact form. \(\displaystyle{\lim_{x\to5^+}{\sqrt{x^225}}}\)
Solution 

video by PatrickJMT 

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\(\displaystyle{\lim_{x\to5^}{\sqrt{x(5x)}}}\)
Problem Statement 

Evaluate the limit, giving your answer in exact form. \(\displaystyle{\lim_{x\to5^}{\sqrt{x(5x)}}}\)
Solution 

video by Krista King Math 

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\(\displaystyle{\lim_{x\to0}{1/x}}\)
Problem Statement 

Evaluate the limit, giving your answer in exact form. \(\displaystyle{\lim_{x\to0}{1/x}}\)
Solution 

video by Khan Academy 

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\(\displaystyle{\lim_{x\to0}{1/x^2}}\)
Problem Statement 

Evaluate the limit, giving your answer in exact form. \(\displaystyle{\lim_{x\to0}{1/x^2}}\)
Solution 

video by Khan Academy 

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Intermediate Problems
Evaluate \(\displaystyle{\lim_{x\to1}{f(x)}}\) for \(\displaystyle{f(x)=\left\{\begin{array}{lr} x+3 & x \leq 1 \\ x^22x & x >1 \end{array}\right.}\)
Problem Statement 

Evaluate \(\displaystyle{\lim_{x\to1}{f(x)}}\) for \(\displaystyle{f(x)=\left\{\begin{array}{lr} x+3 & x \leq 1 \\ x^22x & x >1 \end{array}\right.}\)
Solution 

video by PatrickJMT 

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Prove that the limit \(\displaystyle{\lim_{x\to0}{\frac{x}{x}}}\) does not exist.
Problem Statement 

Prove that the limit \(\displaystyle{\lim_{x\to0}{\frac{x}{x}}}\) does not exist.
Solution 

video by Krista King Math 

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\(\displaystyle{\lim_{x\to0}{\frac{x2\abs{x}}{\abs{x}}}}\)
Problem Statement 

Evaluate the limit, giving your answer in exact form. \(\displaystyle{\lim_{x\to0}{\frac{x2\abs{x}}{\abs{x}}}}\)
Solution 

video by Khan Academy 

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