Limit Laws and Techniques: What is the difference between left and right limits?

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In summary, the difference between x->c^- and x->c^+ is that the former approaches c from the negative side of the x-axis while the latter approaches c from the positive side. When dealing with piecewise functions, the limit only exists if both the left and right limits exist and are equal.
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What is the difference of x->c^- and x->c^+?
 
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\(\displaystyle \lim_{x \to c^-}\) approaches c from the negative side of the x-axis and \(\displaystyle \lim_{x \to c^+}\) approaches c from the positive side. For example \(\displaystyle \lim_{x \to 0^+} \dfrac{1}{x} \to + \infty\) whereas \(\displaystyle \lim_{x \to 0^-} \dfrac{1}{x} \to - \infty\).

-Dan
 
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Or you can have a "piecewise" function like
f(x)= 3x+ 1 if x< 2 and
f(x)= 2x- 2 if x> 2

Then \(\displaystyle \lim_{x\to 2^-} f(x)= \lim_{x\to 2} 3x+ 1= 3(2)+ 1= 7\) since we only look at x less than 2 and
\(\displaystyle \lim_{x\to 2^+} f(x)= \lim_{x\to 2} 2x- 2= 2(2)- 2= 2\) since we only look at x greater than 2.

For a general function, g, the limit, \(\displaystyle \lim_{x\to a} g(x)\) exists if and only if both \(\displaystyle \lim_{x\to 2^-} g(x)\) and \(\displaystyle \lim_{x\to a+} g(x)\) exist and are equal (and, of course, \(\displaystyle \lim_{x\to a} g(x)\) is their common value).
 
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1. What are the basic limit laws?

The basic limit laws include the sum law, difference law, constant multiple law, product law, quotient law, power law, and root law. These laws provide rules for evaluating limits of functions that involve arithmetic operations.

2. How are limit laws used to evaluate limits?

Limit laws are used to break down complex limits into simpler ones that can be evaluated using known techniques. By applying the appropriate limit law, the limit of a function can be rewritten in a simpler form, making it easier to calculate.

3. What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit only considers the behavior of a function from one direction (left or right) of a specific point, while a two-sided limit considers the behavior of a function from both directions. The two-sided limit is used to determine if a function approaches the same value from both sides, while the one-sided limit is used to determine if a function approaches a specific value from one side only.

4. Can limit laws be used to evaluate limits at infinity?

Yes, limit laws can be used to evaluate limits at infinity. In this case, the limit laws are applied to the function as x approaches infinity or negative infinity. For example, the quotient law can be used to evaluate the limit of a rational function at infinity.

5. Are there any limitations to using limit laws to evaluate limits?

Yes, there are certain cases where limit laws cannot be used to evaluate limits. These include limits involving indeterminate forms such as 0/0 or infinity/infinity, as well as limits of functions with discontinuities or infinite oscillations. In these cases, other techniques such as L'Hopital's rule or the squeeze theorem may be used to evaluate the limit.

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