Understanding Vertical Asymptotes: Analyzing Function Behavior as x Approaches Specific Values

Vertical Asymptote at x = -3 and x=4

To find the vertical asymptotes of a function, we need to examine its behavior as x approaches certain values

To find the vertical asymptotes of a function, we need to examine its behavior as x approaches certain values.

For a vertical asymptote at x = -3, we need to check the behavior of the function as x approaches -3 from both the left and the right. Let’s denote the function as f(x).

As x approaches -3 from the left side (x < -3), we can write it as x → -3⁻. We have to evaluate the limit of the function as x approaches -3 from the left side. Symbolically, this can be represented as: lim(x → -3⁻) f(x). Similarly, for a vertical asymptote at x = 4, we need to check the behavior of the function as x approaches 4 from both the left and the right. Let's denote the function as g(x). As x approaches 4 from the left side (x < 4), we can write it as x → 4⁻. We have to evaluate the limit of the function as x approaches 4 from the left side. Symbolically, this can be represented as: lim(x → 4⁻) g(x). To determine the vertical asymptotes, we need to check the following limits: 1. lim(x → -3⁻) f(x) 2. lim(x → -3⁺) f(x) 3. lim(x → 4⁻) g(x) 4. lim(x → 4⁺) g(x) If any of these limits tends to positive or negative infinity, it indicates the presence of a vertical asymptote at that specific x-value. However, since you didn't provide a specific function, we cannot calculate the limits or provide more detailed assistance. Note that the behavior of a function can vary greatly depending on the specific equation or function you want to examine. Please provide additional information or a specific function for us to analyze further and provide a detailed response.

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