Evaluating Continuity of a Piecewise Function at a Boundary
Consider the function \(f(x)\) defined as follows: \[ f(x) = \begin{cases} 2x – 1, & x < 3 \\ kx + 2, & x \ge 3 \end{cases} \] Determine the value of \(k\) that makes \(f(x)\) continuous at \(x = 3\) by ensuring the left-hand limit and right-hand limit at \(x = 3\) are equal.
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 […]
Evaluating One-Sided Limits of a Piecewise Function
Consider the piecewise-defined function \(f(x)\) given by: \[ f(x) = \begin{cases} 2x + 3, & x < 1 \\ x^2, & x \geq 1 \end{cases} \] Solve the following multi-part problem regarding the limits of \(f(x)\) as \(x\) approaches 1.
Continuity Analysis of a Piecewise Function on a Closed Interval
Consider the function f defined on the closed interval [1,4] as follows: For \(1 \le x < 3\), \(f(x) = 2x – 1\). For \(3 \le x \le 4\), \(f(x) = x^2 – 4\). Answer the following: (a) State the definition of continuity at a point. (b) Examine whether the function f is continuous at […]
Assessing One-Sided Limits and Discontinuities
Examine the function \( f(x) \) defined piecewise by: \( f(x) = \begin{cases} \frac{x^2-9}{x-3} & \text{if } x < 3 \\ 5 & \text{if } x \geq 3 \end{cases} \). Answer the following: (a) Evaluate the left-hand and right-hand limits as \(x\) approaches 3 and determine whether the overall limit exists at \(x = 3\). […]
Part a
Consider the function \( f(x) \) defined by: \( f(x) = \begin{cases} \frac{x^2-9}{x-3} & \text{if } x < 3 \\ 5 & \text{if } x \geq 3 \end{cases} \). Evaluate the left-hand limit \(\lim_{x \to 3^-} f(x)\) and the right-hand limit \(\lim_{x \to 3^+} f(x)\). Based on these, determine whether the overall limit of \(f(x)\) […]
Part b
Based on the one-sided limits determined in part (a), identify the type of discontinuity at \(x = 3\) for the function \(f(x)\). Provide a brief justification for your answer.
Part a
For the function \(f(x) = \begin{cases} 2x + 3 & \text{if } x < 1 \\ 5 & \text{if } x \geq 1 \end{cases}\), compute the left-hand limit \(\lim_{x \to 1^-} f(x)\).
Part b
Discuss the significance of one-sided limits in determining the continuity of \(f(x)\) at \(x=1\), based on the result from part (a).
One-Sided Limit Analysis at a Boundary
Consider the piecewise function defined by \(f(x) = \begin{cases} 2x + 3 & \text{if } x < 1 \\ 5 & \text{if } x \geq 1 \end{cases}\). (a) Compute the left-hand limit \(\lim_{x \to 1^-} f(x)\). (b) Discuss the significance of the one-sided limit in determining the continuity of \(f(x)\) at \(x=1\).
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