Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

1. Limits and Continuity

Continuity

Calculus

1. Limits and Continuity

Continuity

Previous problemNext problem

2:38 minutes

Problem 2.6.30

Textbook Question

Determine the interval(s) on which the following functions are continuous.
f(t)=t+2 / t^2−4

Verified step by step guidance

The function $ f(t) = \frac{t+2}{t^2-4} $ is a rational function, which is continuous everywhere in its domain.

A rational function is undefined where its denominator is zero. Set the denominator equal to zero: $ t^2 - 4 = 0 $.

Factor the equation: $ (t-2)(t+2) = 0 $. This gives the solutions $ t = 2 $ and $ t = -2 $.

The function is undefined at $ t = 2 $ and $ t = -2 $. Therefore, these points are not in the domain of the function.

The function is continuous on the intervals $(-\infty, -2)$, $(-2, 2)$, and $(2, \infty)$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Continuity of Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For a function to be continuous over an interval, it must be continuous at every point within that interval. This means there are no breaks, jumps, or asymptotes in the function's graph.

Identifying Discontinuities

Discontinuities in a function can occur due to points where the function is undefined, such as division by zero. For the function f(t) = (t + 2) / (t^2 - 4), we need to find values of t that make the denominator zero, as these will indicate points of discontinuity. Factoring the denominator helps identify these critical points.

Intervals of Continuity

Once discontinuities are identified, the next step is to determine the intervals of continuity. This involves analyzing the real number line and excluding the points of discontinuity to find continuous segments. The function is continuous on intervals that do not include these points, which can be expressed in interval notation.