Examine the continuity of $f(x)=\begin{cases}x+2, & x 1\end{cases}$ at $x=1$.

Examine the continuity of f(x)= x+2, x<1\3, x=1x²+1, x>1 at x=1.

Exercises · Q9

f(x)=\begin{cases}x+2, & x<1\\ 3, & x=1\\ x^{2}+1, & x>1\end{cases}

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Step 1 — Left-hand limit.

\lim_{x\to1^{-}}f(x)=\lim_{x\to1^{-}}(x+2)=1+2=3

Step 2 — Right-hand limit.

\lim_{x\to1^{+}}f(x)=\lim_{x\to1^{+}}(x^{2}+1)=1^{2}+1=2

Step 3 — Value of the function at the point.

f(1)=3 \quad\text{(given directly by the middle branch)}

Step 4 — Apply the continuity test. LHL $(=3)\neq$ RHL $(=2)$ , so $\displaystyle\lim_{x\to1}f(x)$ does not exist at all — meaning condition 2 of continuity fails regardless of what $f(1)$ equals. $f$ is therefore discontinuous at $x=1$ (specifically, a jump discontinuity). …

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Q.Examine the continuity of $f(x)=\begin{cases}x+2, & x<1\\ 3, & x=1\\ x^{2}+1, & x>1\end{cases}$ at $x=1$ .

Concept understanding — Continuity of a Function at a Point

Q.Examine the continuity of f(x)={x+2,x<13,x=1x2+1,x>1f(x)=\begin{cases}x+2, & x<1\\ 3, & x=1\\ x^{2}+1, & x>1\end{cases}f(x)=⎩⎨⎧​x+2,3,x2+1,​x<1x=1x>1​ at x=1x=1x=1.

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Q.Examine the continuity of $f(x)=\begin{cases}x+2, & x<1\\ 3, & x=1\\ x^{2}+1, & x>1\end{cases}$ at $x=1$ .

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