Question

the graph of the function ( f ) is shown. for which of the following values of ( a ) does ( lim_{x \to a} f(x) ) not exist?
options:
a) 1 only
b) 5 only
c) 3 and 5 only
d) 1, 3, and 5
(note: the graph is labeled \graph of ( f )\ with a coordinate plane and a piecewise function plot.)

Explanation:

To determine where \(\lim_{x \to a} f(x)\) does not exist, we check the left - hand limit (LHL) and right - hand limit (RHL) at each \(a\):

For \(a = 1\):

As \(x\) approaches \(1\) from the left and from the right, the graph of \(f(x)\) approaches the same value. So, \(\lim_{x
ightarrow1^{-}}f(x)=\lim_{x
ightarrow1^{+}}f(x)\), and \(\lim_{x
ightarrow1}f(x)\) exists.

For \(a = 3\):

As \(x\) approaches \(3\) from the left and from the right, the graph of \(f(x)\) approaches the same value (the minimum point of the curve). So, \(\lim_{x
ightarrow3^{-}}f(x)=\lim_{x
ightarrow3^{+}}f(x)\), and \(\lim_{x
ightarrow3}f(x)\) exists.

For \(a = 5\):

As \(x\) approaches \(5\) from the left, the graph of \(f(x)\) is approaching a different value than when \(x\) approaches \(5\) from the right (because of the sharp corner or different behavior on either side of \(x = 5\)). So, \(\lim_{x
ightarrow5^{-}}f(x)
eq\lim_{x
ightarrow5^{+}}f(x)\), and \(\lim_{x
ightarrow5}f(x)\) does not exist.

So, the limit does not exist only when \(a = 5\).

Answer:

B. 5 only