Question

1. use the graph of ( y = f(x) ) below to find the following limits.

a. ( lim_{x \to 2^-} f(x) = 1 )
b. ( lim_{x \to 2^+} f(x) = 2 )
c. ( lim_{x \to 2} f(x) = dne )
d. ( lim_{x \to 0^-} f(x) = )
e. ( lim_{x \to 0^+} f(x) )
f. ( lim_{x \to 0} f(x) )

Explanation:

To solve the limit problems using the graph of \( y = f(x) \), we analyze the behavior of the function as \( x \) approaches the given values from the left, right, or both sides.

Part (d): \( \boldsymbol{\lim_{x \to 0^-} f(x)} \)

Step 1: Understand the notation

\( x \to 0^- \) means \( x \) approaches \( 0 \) from the left (negative side).

Step 2: Analyze the graph

As \( x \) approaches \( 0 \) from the left (moving along the graph towards \( x = 0 \) from values like \( x = -1, -0.5, \) etc.), the graph of \( f(x) \) approaches a specific \( y \)-value. From the graph, the left-hand side of \( x = 0 \) follows a line that approaches \( y = -2 \) (or the \( y \)-intercept, which appears to be \( -2 \) based on the grid).

Part (e): \( \boldsymbol{\lim_{x \to 0^+} f(x)} \)

Step 1: Understand the notation

\( x \to 0^+ \) means \( x \) approaches \( 0 \) from the right (positive side).

Step 2: Analyze the graph

As \( x \) approaches \( 0 \) from the right (moving along the graph towards \( x = 0 \) from values like \( x = 0.5, 0.1, \) etc.), the graph of \( f(x) \) also approaches the same \( y \)-value as the left-hand limit (since the graph is continuous near \( x = 0 \)). Thus, it approaches \( y = -2 \).

Part (f): \( \boldsymbol{\lim_{x \to 0} f(x)} \)

Step 1: Recall the limit definition

For \( \lim_{x \to 0} f(x) \) to exist, the left-hand limit \( \lim_{x \to 0^-} f(x) \) and the right-hand limit \( \lim_{x \to 0^+} f(x) \) must be equal.

Step 2: Compare left and right limits

From parts (d) and (e), \( \lim_{x \to 0^-} f(x) = -2 \) and \( \lim_{x \to 0^+} f(x) = -2 \). Since these are equal, \( \lim_{x \to 0} f(x) = -2 \).

Final Answers:

d. \( \boldsymbol{\lim_{x \to 0^-} f(x) = -2} \)
e. \( \boldsymbol{\lim_{x \to 0^+} f(x) = -2} \)
f. \( \boldsymbol{\lim_{x \to 0} f(x) = -2} \)

Answer:

To solve the limit problems using the graph of \( y = f(x) \), we analyze the behavior of the function as \( x \) approaches the given values from the left, right, or both sides.

Part (d): \( \boldsymbol{\lim_{x \to 0^-} f(x)} \)

Step 1: Understand the notation

\( x \to 0^- \) means \( x \) approaches \( 0 \) from the left (negative side).

Step 2: Analyze the graph

As \( x \) approaches \( 0 \) from the left (moving along the graph towards \( x = 0 \) from values like \( x = -1, -0.5, \) etc.), the graph of \( f(x) \) approaches a specific \( y \)-value. From the graph, the left-hand side of \( x = 0 \) follows a line that approaches \( y = -2 \) (or the \( y \)-intercept, which appears to be \( -2 \) based on the grid).

Part (e): \( \boldsymbol{\lim_{x \to 0^+} f(x)} \)

Step 1: Understand the notation

\( x \to 0^+ \) means \( x \) approaches \( 0 \) from the right (positive side).

Step 2: Analyze the graph

As \( x \) approaches \( 0 \) from the right (moving along the graph towards \( x = 0 \) from values like \( x = 0.5, 0.1, \) etc.), the graph of \( f(x) \) also approaches the same \( y \)-value as the left-hand limit (since the graph is continuous near \( x = 0 \)). Thus, it approaches \( y = -2 \).

Part (f): \( \boldsymbol{\lim_{x \to 0} f(x)} \)

Step 1: Recall the limit definition

For \( \lim_{x \to 0} f(x) \) to exist, the left-hand limit \( \lim_{x \to 0^-} f(x) \) and the right-hand limit \( \lim_{x \to 0^+} f(x) \) must be equal.

Step 2: Compare left and right limits

From parts (d) and (e), \( \lim_{x \to 0^-} f(x) = -2 \) and \( \lim_{x \to 0^+} f(x) = -2 \). Since these are equal, \( \lim_{x \to 0} f(x) = -2 \).

Final Answers:

d. \( \boldsymbol{\lim_{x \to 0^-} f(x) = -2} \)
e. \( \boldsymbol{\lim_{x \to 0^+} f(x) = -2} \)
f. \( \boldsymbol{\lim_{x \to 0} f(x) = -2} \)