Identifying and Correcting Limit Law Misapplications in Piecewise Models
Consider the piecewise function \( f(x) \) defined as follows: For \( x < 2 \): \( f(x) = \frac{x^2 – 4}{x – 2} \) For \( x \geq 2 \): \( f(x) = 2x – 2 \) A student attempts to evaluate \( \lim_{x \to 2} f(x) \) by simplifying the expression in the […]
Identifying Misapplications in Piecewise Limits
Consider the piecewise function defined as follows:\n\nFor \(t < 2\), \(f(t) = 2t + 3\)\nFor \(t ≥ 2\), \(f(t) = t² – 1\)\n\nA student computed \(\lim_{t\to 2} f(t)\) by simply evaluating \(f(2)\), thereby misapplying the limit laws.\n\nAnswer the following parts:
Evaluating Complex Limits via Sum and Product Laws
Evaluate the following limits using appropriate limit laws.
Simplifying and Evaluating Limits of Rational Functions
Consider the function \( f(x) = \frac{x^2 – 4}{x – 2} \). (a) Simplify the rational expression by factoring and cancelling common terms. (b) Using your simplification, compute the limit \(\lim_{x \to 2} f(x)\).
Limit Evaluation Using Factoring Techniques
Evaluate the limit: \(\lim_{x \to 2} \frac{x^2 – 4}{x – 2}\).