Techniques for Evaluating One-Sided Limits
One-sided limits can be evaluated using graphical representations or tables of values. When using a graph, one observes the trend of the function as it nears the point from one side to predict the limit. Tables of values provide numerical evidence of the function's behavior as it approaches the point from one side. For example, consider a piecewise function \( f(x) = \{1 \text{ if } x < 2, 3 \text{ if } x \geq 2\} \). The left-sided limit at \( x=2 \), \( \lim_{x \to 2^-} f(x) \), is 1, while the right-sided limit, \( \lim_{x \to 2^+} f(x) \), is 3. This discrepancy indicates that the function does not have an overall limit at \( x=2 \).Identifying Vertical Asymptotes with One-Sided Limits
One-sided limits are also crucial for identifying vertical asymptotes, which are characterized by the function's values increasing without bound as it approaches a certain point. A vertical asymptote at point \( a \) is present if the left-sided limit, the right-sided limit, or both diverge to infinity or negative infinity as \( x \) approaches \( a \). For instance, the function \( f(x) = \frac{1}{x} \) exhibits a vertical asymptote at \( x=0 \), with the left-sided limit tending towards negative infinity and the right-sided limit tending towards positive infinity. This behavior indicates that the function's values become unbounded in opposite directions as \( x \) approaches zero from the left and right.Practical Implications and Summary of One-Sided Limits
Real-world examples of one-sided limits can help clarify their practical significance. Take the function \( f(x) = \frac{|x|}{x} \), which is undefined at \( x=0 \). The left-sided limit at \( x=0 \) is -1, as \( x \) approaches zero from negative values, while the right-sided limit is 1, as \( x \) approaches zero from positive values. This demonstrates that the overall limit at \( x=0 \) does not exist. In conclusion, one-sided limits are an essential tool in calculus for analyzing the behavior of functions near specific points, determining the existence of limits, and identifying points of discontinuity or vertical asymptotes.
