To determine the limit of a piecewise function, we need to analyze the function's behavior as it approaches a particular point, considering each piece of the function that might be relevant near that point. Here’s a step-by-step guide to finding limits for piecewise functions:

Step-by-Step Approach for Finding Limits of Piecewise Functions

1. Identify the point at which the limit is being evaluated:

   • Let’s denote this point as $x = a$. The value of $a$ is where you want to find the limit of the piecewise function.

2. Examine the definition of the piecewise function:

   • Write down the different pieces of the function and their corresponding intervals. For example, a piecewise function might look like this: \begin{cases} f_1(x) & \text{if } x < a \\ f_2(x) & \text{if } x = a \\ f_3(x) & \text{if } x > a \end{cases}$$

3. Determine the left-hand limit ($\lim_{x \to a^-} f(x)$):

   • Use the piece of the function defined for values less than $a$ (typically $f_1(x)$). Substitute values approaching $a$ from the left into this piece and find the limit: $\lim_{x \to a^-} f(x) = \lim_{x \to a^-} f_1(x)$
   • This involves plugging in values just slightly less than $a$ into $f_1(x)$.

4. Determine the right-hand limit ($\lim_{x \to a^+} f(x)$):

   • Use the piece of the function defined for values greater than $a$ (typically $f_3(x)$). Substitute values approaching $a$ from the right into this piece and find the limit: $\lim_{x \to a^+} f(x) = \lim_{x \to a^+} f_3(x)$
   • This involves plugging in values just slightly greater than $a$ into $f_3(x)$.

5. Compare the left-hand and right-hand limits:

   • If $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$ are equal, then the two-sided limit exists and is equal to that common value: $\lim_{x \to a} f(x) = \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$
   • If the left-hand limit and right-hand limit are not equal, then the limit does not exist at $x = a$.

Example:

Suppose we have the piecewise function:

\begin{cases} 2x + 1 & \text{if } x < 3 \\ 5 & \text{if } x = 3 \\ x^2 - 4 & \text{if } x > 3 \end{cases}$$ Find $$\lim_{x \to 3} f(x)$$. **Step 1**: $$a = 3$$. **Step 2**: Identify the relevant pieces of the function: - For $$x < 3$$, use $$f(x) = 2x + 1$$. - For $$x = 3$$, use $$f(x) = 5$$. - For $$x > 3$$, use $$f(x) = x^2 - 4$$. **Step 3**: Find the left-hand limit: $$\lim_{x \to 3^-} (2x + 1) = 2(3) + 1 = 7$$ **Step 4**: Find the right-hand limit: $$\lim_{x \to 3^+} (x^2 - 4) = 3^2 - 4 = 5$$ **Step 5**: Compare the limits: - $$\lim_{x \to 3^-} f(x) = 7$$ - $$\lim_{x \to 3^+} f(x) = 5$$ Since these two limits are not equal, $$\lim_{x \to 3} f(x)$$ **does not exist**. Would you like more details or have any questions about this process? ### 5 Related Questions: 1. How do you find limits at points of discontinuity in a piecewise function? 2. What are one-sided limits, and why are they important in evaluating piecewise functions? 3. How can you determine if a piecewise function is continuous at a point? 4. What is the difference between jump discontinuity and removable discontinuity? 5. How can graphing a piecewise function help in finding its limit? ### Tip: When finding limits of piecewise functions, always pay close attention to which piece is defined on the interval approaching the point, as it dictates which formula you use for left- and right-hand limits.