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lim{x->7} (x-7)/(x^2-8x+7)

Check the final answer first, then review the worked steps.

Problem

lim_{x->7} (x-7)/(x^2-8x+7)

Answer

0.1667

Step-by-step solution

  1. Analyze the function: The given function is $f(x) = \frac{x - 7}{x^2 - 8x + 7}$. We want to find the limit as $x$ approaches 7.
  1. Simplify the expression: Factor the denominator $x^2 - 8x + 7$. We look for two numbers that multiply to 7 and add to -8, which are -1 and -7. Thus, $x^2 - 8x + 7 = (x - 1)(x - 7)$. The function becomes $f(x) = \frac{x - 7}{(x - 1)(x - 7)}$.
  1. Cancel common terms: For $x \neq 7$, we can cancel the $(x - 7)$ terms: $f(x) = \frac{1}{x - 1}$.
  1. Evaluate the limit: Now we can evaluate the limit by direct substitution into the simplified expression: $\lim_{x \to 7} \frac{1}{x - 1} = \frac{1}{7 - 1} = \frac{1}{6}$.
  1. Convert to decimal: The value $\frac{1}{6}$ is approximately $0.166666...$. Rounding to four decimal places as requested, we get $0.1667$. This confirms the value that should be placed in the table for $x=7$ and the final limit estimate.