Question
How to evaluate one-sided limits from a piecewise function graph
Original question: Graph the following piecewise function, then evaluate each limit below. $$f(x)=\begin{cases} -x+4 & \text{if } x\le 3\\ x-5 & \text{if } x>3 \end{cases}$$ a. $$\lim_{x\to 3^-} f(x)=1$$ b. $$\lim_{x\to 3^+} f(x)=-2$$ c. $$\lim_{x\to 3} f(x)$$ does not exist
Expert Verified Solution
Expert intro: A piecewise graph is really two stories meeting at one point. The key is to use the rule that matches the side you are approaching from, not the rule that looks nicest.
Detailed walkthrough
We read the function by side of approach:
-x+4 & \text{if } x\le 3\\ x-5 & \text{if } x>3 \end{cases}$$ ### a. Left-hand limit As $x\to 3^-$, we stay on the branch $f(x)=-x+4$. $$\lim_{x\to 3^-} f(x)=-(3)+4=1$$ ### b. Right-hand limit As $x\to 3^+$, we use $f(x)=x-5$. $$\lim_{x\to 3^+} f(x)=3-5=-2$$ ### c. Two-sided limit A two-sided limit exists only if both one-sided limits match. Here, $$\lim_{x\to 3^-} f(x)=1 \neq -2=\lim_{x\to 3^+} f(x)$$ So, $$\lim_{x\to 3} f(x) \text{ does not exist}$$ ### Quick check If you also wanted the function value at the breakpoint, then because $x\le 3$ uses the first rule, $$f(3)=-3+4=1$$ That is the value of the function, not the two-sided limit. ### š” Pitfall guide A common mistake is plugging $x=3$ into the wrong branch. For one-sided limits, always match the side: left uses $x<3$, right uses $x>3$. Another trap is thinking the function value must equal the limit; they can be different. ### š Real-world variant If the breakpoint were changed from $x=3$ to another value, the method stays the same: choose the formula on the left or right side, then substitute the approach value into that expression. If both sides happened to give the same number, the two-sided limit would exist even if the function definition at the point were different. ### š Related terms piecewise function, one-sided limit, jump discontinuityFAQ
How do you find one-sided limits from a piecewise function?
Use the formula that applies on the side you are approaching from. For xāaā, use the branch valid for x less than a. For xāa+, use the branch valid for x greater than a.
When does a two-sided limit fail to exist?
A two-sided limit fails to exist when the left-hand and right-hand limits are different. If they match, the limit exists and equals that common value.