Solve: lim_(x rarr4)(x)/(x-4)

Identify Function & Point: Identify the function and the point of interest. We are looking at the limit of the function f(x) = x/(x-4) as x approaches 4. Substitute x = 4: Substitute x = 4 directly into the function to check for any immediate simplification or issues. f(4) = 4/(4-4) = 4/0, which is undefined, indicating a potential issue at x = 4. Recognize Discontinuity: Recognize the type of discontinuity. Since direct substitution leads to a division by zero, this suggests a vertical asymptote at x = 4, meaning the limit might be infinite or does not exist. Analyze Behavior Approaching 4: Analyze the behavior of the function as x approaches 4 from both sides (left and right limits). As x approaches 4 from the left (x -> 4^-), f(x) = x/(x-4) -> -∞. As x approaches 4 from the right (x -> 4^+), f(x) = x/(x-4) -> +∞. Conclude Limit: Conclude the limit based on the behavior from both sides. Since the left-hand limit and the right-hand limit are not equal and both approach infinity but with opposite signs, the overall limit does not exist.

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Find derivatives using the quotient rule I

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Q. Solve:

\lim_{x \rightarrow 4}\frac{x}{x-4}

Identify Function & Point: Identify the function and the point of interest. We are looking at the limit of the function $f(x) = \frac{x}{x-4}$ as $x$ approaches $4$ .
Substitute $x = 4$ : Substitute $x = 4$ directly into the function to check for any immediate simplification or issues. $f(4) = \frac{4}{(4-4)} = \frac{4}{0}$ , which is undefined, indicating a potential issue at $x = 4$ .
Recognize Discontinuity: Recognize the type of discontinuity. Since direct substitution leads to a division by $0$ , this suggests a vertical asymptote at $x = 4$ , meaning the limit might be infinite or does not exist.
Analyze Behavior Approaching $4$ : Analyze the behavior of the function as $x$ approaches $4$ from both sides (left and right limits). As $x$ approaches $4$ from the left ( $x \to 4^-$ ), $f(x) = \frac{x}{x-4} \to -\infty$ . As $x$ approaches $4$ from the right ( $x \to 4^+$ ), $f(x) = \frac{x}{x-4} \to +\infty$ .
Conclude Limit: Conclude the limit based on the behavior from both sides. Since the left-hand limit and the right-hand limit are not equal and both approach $\infty$ but with opposite signs, the overall limit does not exist.