Limits and Continuity Worksheet

Limits and Continuity Worksheet

Read each question carefully and provide your answer in the space provided. Show all your work for full credit.

1. What is the value of \(\lim_{x \to 2} (3x^2 - 5x + 1)\)?

2. For a function to be continuous at a point c, which of the following conditions must be met?

3. A function is said to be   at a point if the limit of the function as x approaches that point exists, the function is defined at that point, and the limit equals the function's value at that point.

4. The limit of \(f(x) = \frac{x^2 - 4}{x - 2}\) as x approaches 2 is  .

5. Evaluate the limit: \(\lim_{x \to 0} \frac{\sin(x)}{x}\).

6. Determine if the function \(f(x) = \begin{cases} x^2+1, & x < 1 \cr 3x-1, & x \ge 1 \end{cases}\) is continuous at \(x=1\). Show your work.

7. If a function is differentiable at a point, then it must be continuous at that point.

8. A removable discontinuity occurs when the left-hand limit and the right-hand limit are equal but not equal to the function's value at that point, or the function is undefined at that point.

9. Consider the graph of a function f(x) below. Based on the graph, answer the following questions:

a) \(\lim_{x \to 2^-} f(x) = \)

b) \(\lim_{x \to 2^+} f(x) = \)

c) Is f(x) continuous at \(x=2\)? Explain.