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To determine if a specific x-value results in a vertical asymptote or a hole in a function, you need to consider the behavior of the function as it approaches that x-value from both sides.
1. Vertical Asymptote: If the function approaches positive or negative infinity as x approaches the specific x-value, then there is a vertical asymptote at that x-value. This means that the function cannot be evaluated at that x-value, and the graph will have a vertical line that it approaches but never touches.
2. Hole: If the function approaches a finite value as x approaches the specific x-value, then there is a hole at that x-value. This means that the function can be evaluated at that x-value, but there is a discontinuity in the graph at that point. The graph will have an open circle or a "hole" at that x-value.
To determine if a specific x-value results in a vertical asymptote or a hole, you can analyze the function algebraically by factoring or simplifying the expression. Look for any common factors that cancel out and create a hole, or any factors in the denominator that result in division by zero and create a vertical asymptote.
It is important to note that graphing can provide a visual confirmation of the presence of a vertical asymptote or a hole, but it is not the only method of determining their existence. Analyzing the behavior of the function as x approaches the specific x-value can give you valuable information about the presence of vertical asymptotes or holes without graphing the entire function.
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