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Multiple Choice
Use the graph of f(x)to estimate the value of the limit or state that it does not exist (DNE). limx→0f(x)
A
0
B
1
C
5
D
DNE
Verified step by step guidance
1
Step 1: Understand the problem. We are tasked with estimating the value of the limit \( \lim_{x \to 0} f(x) \) using the graph of \( f(x) \). The limit represents the behavior of the function \( f(x) \) as \( x \) approaches 0 from both the left and the right.
Step 2: Analyze the graph near \( x = 0 \). Observe the behavior of the function \( f(x) \) as \( x \) approaches 0 from the left (negative side) and from the right (positive side).
Step 3: From the graph, note that as \( x \to 0^- \) (approaching 0 from the left), the function \( f(x) \) increases without bound, heading towards positive infinity. Similarly, as \( x \to 0^+ \) (approaching 0 from the right), \( f(x) \) also increases without bound, heading towards positive infinity.
Step 4: Recall the definition of a limit. For the limit \( \lim_{x \to 0} f(x) \) to exist, the function must approach a single finite value from both sides as \( x \to 0 \). In this case, \( f(x) \) does not approach a finite value; instead, it diverges to infinity.
Step 5: Conclude that the limit \( \lim_{x \to 0} f(x) \) does not exist (DNE) because the function diverges to infinity as \( x \to 0 \).