Worked problem · Calculus I · Unit 1

One Sided Piecewise Limit: Worked Example and Solution

A complete one-sided branch analysis example with method choice, derivation, verification, and a common wrong approach.

1 problem8 min estimated timeFoundational to intermediate progression

Problem

For $f(x)=\begin{cases}x+2,&x<1\\4-x,&x\ge1\end{cases}$ , evaluate the limit as $x\to1^-$ .

Complete worked solutions

Every problem has a source-matched answer and independently reviewed derivation.

Problem 1: For $f(x)=\begin{cases}x+2,&x<1\\4-x,&x\ge1\end{cases}$ , evaluate the limit as $x\to1^-$ .

Answer: $3$

Why this method: One-sided branch analysis matches the mathematical structure before any algebraic cleanup.

Approaching from the left selects the branch $f(x)=x+2$ .
As $x\to1^-$ , $x+2\to3$ ; the value assigned by the other branch at the endpoint does not control this one-sided limit.
Therefore the result is $3$ .

What is included

See a concise answer and full derivation for for f (x) equals piecewise function: row 1: x plus 2,, x less than 1; row 2: 4 minus x,, x greater than or equal to 1, evaluate the limit as x approaches 1 to the power (minus).

Skills assessed

One-sided branch analysis

Prerequisites

function notation
algebraic factoring

Common errors

Using the branch selected by the endpoint equality instead of the approach direction.

Problem

Complete worked solutions

Problem 1: For f(x)={x+2,x<14−x,x≥1f(x)=\begin{cases}x+2,&x<1\\4-x,&x\ge1\end{cases}f(x)={x+2,4−x,​x<1x≥1​, evaluate the limit as x→1−x\to1^-x→1−.

What is included

Skills assessed

Prerequisites

Common errors

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Supporting steps

Problem 1: For $f(x)=\begin{cases}x+2,&x<1\\4-x,&x\ge1\end{cases}$ , evaluate the limit as $x\to1^-$ .