Continuity and one-sided limits

The left-hand limit $\lim_{x\to a^-}f(x)$ is the value approached as $x$ comes toward $a$ from below, and the right-hand limit $\lim_{x\to a^+}f(x)$ is the value approached from above. For a piecewise function, use the $x<a$ piece for the left-hand limit and the $x\ge a$ piece for the right-hand limit.

When all three agree, $f$ is continuous at $x=a$. For example, if $f(x)=2x$ for $x<4$ and $f(x)=x+a$ for $x\ge 4$, the left-hand limit $8$ equals the right-hand limit $4+a$ when $a=4$, so that is the value that makes $f$ continuous at $x=4$.

Examples

1. 01What is the left-hand limit at x=1?$\displaystyle f(x)=\begin{cases} 4x & (x<1) \\ x+2 & (x \geq 1) \end{cases}$AnswerHide

   Answer$\displaystyle 4$

   Steps

   1. Coming from the left, use the x<1 branch
   2. $\displaystyle 4 \cdot 1$$\displaystyle = 4$
2. 02Which value of a makes f continuous at x=4?$\displaystyle f(x)=\begin{cases} 2x & (x<4) \\ x+a & (x \geq 4) \end{cases}$AnswerHide

   Answer$\displaystyle 4$

   Steps

   1. Value coming from the left
   2. $\displaystyle 2 \cdot 4$$\displaystyle = 8$
   3. Value coming from the right
   4. $\displaystyle 4+a$
   5. Continuity means these two are equal
   6. $\displaystyle 4+a = 8$$\displaystyle \Rightarrow a = 4$
3. 03What is the right-hand limit at x=4?$\displaystyle f(x)=\begin{cases} 3x & (x<4) \\ x+2 & (x \geq 4) \end{cases}$AnswerHide

   Answer$\displaystyle 6$

   Steps

   1. Coming from the right, use the x≥4 branch
   2. $\displaystyle 4+2$$\displaystyle = 6$
   3. From the left it is the other branch, giving 12
4. 04What is the left-hand limit at x=2?$\displaystyle f(x)=\begin{cases} 4x & (x<2) \\ x-2 & (x \geq 2) \end{cases}$AnswerHide

   Answer$\displaystyle 8$

   Steps

   1. Coming from the left, use the x<2 branch
   2. $\displaystyle 4 \cdot 2$$\displaystyle = 8$
5. 05What is the right-hand limit at x=1?$\displaystyle f(x)=\begin{cases} 3x & (x<1) \\ x-2 & (x \geq 1) \end{cases}$AnswerHide

   Answer$\displaystyle -1$

   Steps

   1. Coming from the right, use the x≥1 branch
   2. $\displaystyle 1-2$$\displaystyle = -1$
   3. From the left it is the other branch, giving 3
6. 06What is the left-hand limit at x=4?$\displaystyle f(x)=\begin{cases} 2x & (x<4) \\ x-1 & (x \geq 4) \end{cases}$AnswerHide

   Answer$\displaystyle 8$

   Steps

   1. Coming from the left, use the x<4 branch
   2. $\displaystyle 2 \cdot 4$$\displaystyle = 8$

FAQ

Q1If the left-hand and right-hand limits agree, is the function continuous?AnswerHide

A

Not yet. The limit must also equal $f(a)$. For example, define $f(x)=x$ for $x\ne 1$ and $f(1)=3$. Both one-sided limits at $x=1$ are $1$, but $f(1)=3$, so $f$ is not continuous at $x=1$.

Q2Which piece do I use to compute $f(a)$?AnswerHide

A

The piece whose inequality includes the equal sign. If the $x\ge a$ piece has it, substitute $x=a$ there. Then the right-hand limit equals $f(a)$, and you only need to check whether the left-hand limit matches.