Question

find each x-value at which f is discontinuous and for each x-value, determine whether f is continuous from the right, or from the left, or neither.
$f(x) = \

$$\begin{cases} x + 5 & \\text{if } x < 0 \\\\ e^x & \\text{if } 0 \\leq x \\leq 1 \\\\ 3 - x & \\text{if } x > 1 \\end{cases}$$

$
$x = \boxed{}$ (smaller value)
is $f$ continuous from the right, left, or neither at this value?
$\circ$ continuous from the right
$\circ$ continuous from the left
$\circ$ neither
$x = \boxed{}$ (larger value)
is $f$ continuous from the right, left, or neither at this value?
$\circ$ continuous from the right
$\circ$ continuous from the left
$\circ$ neither
sketch the graph of $f$.

Explanation:

Step1: Check discontinuity at \( x = 0 \)

• Left - hand limit (\( x \to 0^- \)): For \( x<0 \), \( f(x)=x + 5 \). So, \( \lim_{x\to0^-}f(x)=\lim_{x\to0^-}(x + 5)=0 + 5 = 5 \).
• Right - hand limit (\( x \to 0^+ \)): For \( 0\leq x\leq1 \), \( f(x)=e^x \). So, \( \lim_{x\to0^+}f(x)=\lim_{x\to0^+}e^x=e^0 = 1 \).
• \( f(0)=e^0 = 1 \) (since \( 0\in[0,1] \)).
• Since \( \lim_{x\to0^-}f(x)=5

eq\lim_{x\to0^+}f(x)=1 \), \( f(x) \) is discontinuous at \( x = 0 \).

• Now, check continuity from the right: \( \lim_{x\to0^+}f(x)=f(0) = 1 \), so \( f(x) \) is continuous from the right at \( x = 0 \).

Step2: Check discontinuity at \( x = 1 \)

• Left - hand limit (\( x \to 1^- \)): For \( 0\leq x\leq1 \), \( f(x)=e^x \). So, \( \lim_{x\to1^-}f(x)=\lim_{x\to1^-}e^x=e^1 = e \).
• Right - hand limit (\( x \to 1^+ \)): For \( x>1 \), \( f(x)=3 - x \). So, \( \lim_{x\to1^+}f(x)=\lim_{x\to1^+}(3 - x)=3 - 1 = 2 \).
• \( f(1)=e^1 = e \) (since \( 1\in[0,1] \)).
• Since \( \lim_{x\to1^-}f(x)=e

eq\lim_{x\to1^+}f(x)=2 \), \( f(x) \) is discontinuous at \( x = 1 \).

• Now, check continuity from the left: \( \lim_{x\to1^-}f(x)=f(1) = e \), so \( f(x) \) is continuous from the left at \( x = 1 \).

Answer:

• For the smaller value (\( x = 0 \)):
• \( x=\boldsymbol{0} \)
• Is \( f \) continuous from the right, left, or neither at this value? \( \boldsymbol{\text{continuous from the right}} \)
• For the larger value (\( x = 1 \)):
• \( x=\boldsymbol{1} \)
• Is \( f \) continuous from the right, left, or neither at this value? \( \boldsymbol{\text{continuous from the left}} \)