Exercise Questions

All questions in this exercise are listed below. Click on a question to view its solution.

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Key Concepts

This exercise focuses on the following concepts:

One-Sided Limits

For a piecewise function, the Left-Hand Limit (LHL) and Right-Hand Limit (RHL) are evaluated separately:

${\lim }_{x\to c^{-}}f(x)\text{and}{\lim }_{x\to c^{+}}f(x)$

The two-sided limit exists if and only if LHL = RHL:

${\lim }_{x\to c}f(x)=L⟺{\lim }_{x\to c^{-}}f(x)={\lim }_{x\to c^{+}}f(x)=L$

Continuity at a Point

A function $f(x)$ is continuous at $x=c$ if all three conditions hold:

1. $f(c)$ is defined.
2. ${\lim }_{x\to c}f(x)$ exists (LHL = RHL).
3. ${\lim }_{x\to c}f(x)=f(c)$.

If any one condition fails, the function is discontinuous at $x=c$.

Finding Unknown Constants for Continuity

When a piecewise function contains an unknown constant $k$ (or $m$, $n$), and the function must be continuous at $x=c$:

Step 1: Compute LHL and RHL using the appropriate pieces of the function.

Step 2: Set LHL = RHL to ensure the limit exists.

Step 3: Set the limit equal to $f(c)$ to satisfy the third continuity condition.

Step 4: Solve the resulting equation for the unknown constant.

If continuity is required at two points with two unknowns $m$ and $n$, apply the above steps at each point to form a system of two equations, then solve simultaneously.

Points of Discontinuity

A function fails to be continuous at $x=c$ if:

• $f(c)$ is undefined (e.g., division by zero, square root of a negative number), or
• LHL $\ne$ RHL (the limit does not exist), or
• The limit exists but ${\lim }_{x\to c}f(x)\ne f(c)$.

Identifying these failures helps locate all points of discontinuity in the domain.

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Important Formulas

• Existence of a Limit: ${\lim }_{x\to c}f(x)=L⟺{\lim }_{x\to c^{-}}f(x)={\lim }_{x\to c^{+}}f(x)=L$

• Definition of Continuity — A function $f$ is continuous at $c$ if:

  1. $f(c)$ is defined.
  2. ${\lim }_{x\to c}f(x)$ exists.
  3. ${\lim }_{x\to c}f(x)=f(c)$.

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Summary

This exercise covers the fundamental principles of limits and continuity. Key learnings include evaluating one-sided limits for piecewise functions and applying the three-step continuity test. Mastery involves correctly setting LHL equal to RHL to solve for unknown constants and identifying domain restrictions that lead to discontinuities.