limits and continuity practice problems

📚 Topic Summary

Limits help us understand what value a function approaches as its input gets closer and closer to a certain point. Continuity, on the other hand, means that a function has no breaks or jumps; you can draw its graph without lifting your pen. Many problems involve finding limits and determining whether a function is continuous at a given point. We often use algebraic techniques or graphical analysis to solve them.

🧠 Part A: Vocabulary

Match the following terms with their definitions:

1. Limit
2. Continuity
3. Removable Discontinuity
4. Infinite Discontinuity
5. Piecewise Function

1. A function composed of multiple sub-functions defined on different intervals.
2. A point at which a function is not continuous but can be made continuous by redefining the function at that point.
3. The value that a function approaches as the input approaches a certain value.
4. A condition where a function has no abrupt changes in value; you can draw it without lifting your pen.
5. A discontinuity where the function approaches infinity.

Answer Key:

• Limit - The value that a function approaches as the input approaches a certain value.
• Continuity - A condition where a function has no abrupt changes in value; you can draw it without lifting your pen.
• Removable Discontinuity - A point at which a function is not continuous but can be made continuous by redefining the function at that point.
• Infinite Discontinuity - A discontinuity where the function approaches infinity.
• Piecewise Function - A function composed of multiple sub-functions defined on different intervals.

✏️ Part B: Fill in the Blanks

Complete the following sentences:

1. The limit of a function $f(x)$ as $x$ approaches $a$ exists if the __________ limit and the __________ limit are equal.
2. A function $f(x)$ is continuous at $x=c$ if $\lim_{x \to c} f(x)$ __________, $f(c)$ __________, and $\lim_{x \to c} f(x) = f(c)$.

Answer Key:

• left-hand, right-hand
• exists, is defined

🤔 Part C: Critical Thinking

Explain, in your own words, why understanding limits is essential for understanding continuity. Provide an example of a function where the limit exists at a point, but the function is not continuous at that point.

Example Answer:

Understanding limits is crucial for understanding continuity because continuity requires the limit to exist at a point, and for the function's value at that point to be equal to the limit. Without the limit existing, we can't even begin to discuss continuity. Consider the function:

At $x = 0$, the limit $\lim_{x \to 0} f(x) = 1$, but $f(0) = 2$. Therefore, the limit exists, but the function is not continuous at $x = 0$ because the limit does not equal the function's value at that point.