Continuity, Discontinuities & Asymptotes Assessment

Choose $k$ so $f(x)=x^2$ for $x<2$ and $kx+1$ for $x\ge 2$ is continuous. Which conclusion is correct?

Find the vertical asymptote of $1/(x-5)$. Which conclusion is correct?

Find the horizontal asymptote of $(3x^2+1)/(2x^2-7)$. Which conclusion is correct?

Before attempting worked case A, which statement correctly records the condition that makes ${\lim }_{x\to a}f(x)=f(a)$ usable?

Use ${\lim }_{x\to a}f(x)=f(a)$ to write the first valid mathematical step for case A: Choose $k$ so $f(x)=x^2$ for $x<2$ and $kx+1$ for $x\ge 2$ is continuous.

Carry out the essential calculation for case B: Find the vertical asymptote of $1/(x-5)$. Show at least one meaningful line before the conclusion.

For case C, Find the horizontal asymptote of $(3x^2+1)/(2x^2-7)$. Give the conclusion and one sentence explaining why it follows.

working sort the statements about case A into “supports a valid solution” and “does not support a valid solution”. Case A is: Choose $k$ so $f(x)=x^2$ for $x<2$ and $kx+1$ for $x\ge 2$ is continuous.

Solve case A completely: Choose $k$ so $f(x)=x^2$ for $x<2$ and $kx+1$ for $x\ge 2$ is continuous. State the relationship or method, show the calculation, then check the condition.

Solve case B completely: Find the vertical asymptote of $1/(x-5)$. Keep the key assumption, restriction, or algorithm rule visible in your working.

Solve case C completely: Find the horizontal asymptote of $(3x^2+1)/(2x^2-7)$. Include an exact form, interval, unit, or interpretation whenever the task needs one.

A learner reports “Left limit $4$; $2k+1=4$, so $k=3/2$.” for case A but gives no justification. Which addition makes the response mathematically defensible?

For the three lesson cases below, sort each statement as a “direct conclusion”, a “required check”, or “not justified by the stated mathematics”.

Put a defensible solution path for case A in order. The case is: Choose $k$ so $f(x)=x^2$ for $x<2$ and $kx+1$ for $x\ge 2$ is continuous.

Put a defensible solution path for case B in order. The case is: Find the vertical asymptote of $1/(x-5)$.

Compare the three distinct Continuity, Discontinuities & Asymptotes cases in three labelled mini-responses: A (Choose $k$ so $f(x)=x^2$ for $x<2$ and $kx+1$ for $x\ge 2$ is continuous.), B (Find the vertical asymptote of $1/(x-5)$.), and C (Find the horizontal asymptote of $(3x^2+1)/(2x^2-7)$.). For each, state the representation or first step, the conclusion, and one check; use mathematics rather than a general study tip.

A student begins case B — Find the vertical asymptote of $1/(x-5)$. — by treating this as optional: “For a piecewise function, calculate both one-sided limits and the stated function value. A horizontal asymptote can be crossed at finite $x$.”. Diagnose the mathematical problem and repair the first two lines of the solution.

Solve case B (Find the vertical asymptote of $1/(x-5)$.); Check the method against ${\lim }_{x\to a}f(x)=f(a)$ and For a piecewise function, calculate both one-sided limits and the stated function value. A horizontal asymptote can be crossed at finite $x$. Write a complete solution, including one check that could expose an error.

The lesson requires you to: Classify discontinuities and define continuity at a point and over an interval. Use case C (Find the horizontal asymptote of $(3x^2+1)/(2x^2-7)$.) to explain how the relationship ${\lim }_{x\to a}f(x)=f(a)$ makes that capability possible.

For each case, give the first mathematical step and conclusion/check: (A: Choose $k$ so $f(x)=x^2$ for $x<2$ and $kx+1$ for $x\ge 2$ is continuous.; (B: Find the vertical asymptote of $1/(x-5)$.; (C: Find the horizontal asymptote of $(3x^2+1)/(2x^2-7)$.) in Continuity, Discontinuities & Asymptotes.

For case C — Find the horizontal asymptote of $(3x^2+1)/(2x^2-7)$. — which starting representation or method is best aligned with this lesson?

Analyse the boundary of case A: Choose $k$ so $f(x)=x^2$ for $x<2$ and $kx+1$ for $x\ge 2$ is continuous. Keep its mathematical structure, then specify one changed value, sign, interval, or algorithm decision that would make the condition “For a piecewise function, calculate both one-sided limits and the stated function value. A horizontal asymptote can be crossed at finite $x$.” fail. Decide with working whether the original conclusion Left limit $4$; $2k+1=4$, so $k=3/2$. still follows.

starting from case B — Find the vertical asymptote of $1/(x-5)$. — create one fully specified new numerical or symbolic instance of the same Continuity, Discontinuities & Asymptotes structure. Solve it, check For a piecewise function, calculate both one-sided limits and the stated function value. A horizontal asymptote can be crossed at finite $x$., and explain how its conclusion compares with $x=5$.

Write a proof-quality validation of case C: Find the horizontal asymptote of $(3x^2+1)/(2x^2-7)$. Your answer must show why $y=3/2$. follows from ${\lim }_{x\to a}f(x)=f(a)$, identify a tempting invalid move, and use For a piecewise function, calculate both one-sided limits and the stated function value. A horizontal asymptote can be crossed at finite $x$. to rule it out.

Create an annotated canvas solution for case A: Choose $k$ so $f(x)=x^2$ for $x<2$ and $kx+1$ for $x\ge 2$ is continuous. Show the governing relationship, each key calculation or construction, the condition, and the checked conclusion.

Create an annotated canvas solution for case B: Find the vertical asymptote of $1/(x-5)$. Use a diagram, graph, equation layout, table, or algorithm trace only where it clarifies the specific mathematics.

Create an annotated canvas solution for case C: Find the horizontal asymptote of $(3x^2+1)/(2x^2-7)$. Make the exact answer, interval, unit, direction, or interpretation visible where the case requires it.

The later lesson capability is: Determine values that remove a discontinuity. Compare cases A and B (Choose $k$ so $f(x)=x^2$ for $x<2$ and $kx+1$ for $x\ge 2$ is continuous.; Find the vertical asymptote of $1/(x-5)$.) and decide which case better demonstrates that capability. Support your decision using both actual conclusions (Left limit $4$; $2k+1=4$, so $k=3/2$.; $x=5$.) and the condition For a piecewise function, calculate both one-sided limits and the stated function value. A horizontal asymptote can be crossed at finite $x$.

Act as an examiner auditing three candidate claims about Continuity, Discontinuities & Asymptotes. Sort them into “earns credit”, “earns partial credit only”, and “does not earn credit”; use the concrete worked cases and the stated condition.

build and solve a four-stage Continuity, Discontinuities & Asymptotes problem that deliberately combines the lesson ideas in cases A, B, and C. Use precise values or symbolic data, state how it addresses “Connect infinite limits with vertical asymptotes and limits at infinity with horizontal asymptotes.”, apply ${\lim }_{x\to a}f(x)=f(a)$, and end by checking For a piecewise function, calculate both one-sided limits and the stated function value. A horizontal asymptote can be crossed at finite $x$.

Let $f(x)=\{\begin{array}{ll}2x+1,& x<1,\\ c,& x=1,\\ x+2,& x>1.\end{array}$ Which $c$ makes $f$ continuous at 1?