Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. If $\lim _{x \rightarrow a} f(x) \neq f(a)$ and $\lim _{x \rightarrow a} f(x)$ exists, I can redefine $f(a)$ to make $f$ continuous at $a$.

Name: Problem 54, Chapter 11: Precalculus
Uploaded: 2021-09-11T17:44:18-08:00
Duration: 1 min 10 s
Channel: Farnood Ensan
Description: Determine whether each statement makes sense or does not make sense, and explain your reasoning. If limx →a f(x) ≠f(a) and limx →a f(x) exists, I can redefine f(a) to make f continuous at a.

   Determine whether each statement makes sense or does not make sense, and explain your reasoning.
If $\lim _{x \rightarrow a} f(x) \neq f(a)$ and $\lim _{x \rightarrow a} f(x)$ exists, I can redefine $f(a)$ to make $f$ continuous at $a$.

Precalculus

Robert Blitzer 6th Edition

Chapter 11, Problem 54

Instant Answer

Solved by Expert Farnood Ensan

09/11/2021

Step 1

This means that as $x$ approaches $a$, the function $f(x)$ approaches a certain value, but this value is not equal to $f(a)$. Show more…

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Determine whether each statement makes sense or does not make sense, and explain your reasoning. If $\lim _{x \rightarrow a} f(x) \neq f(a)$ and $\lim _{x \rightarrow a} f(x)$ exists, I can redefine $f(a)$ to make $f$ continuous at $a$.

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Key Concepts

Removable Discontinuity

A removable discontinuity occurs when a function's limit exists as x approaches a certain point, but the function's value at that point is different or undefined. By redefining the function value at this point to equal the limit, the discontinuity is 'removed', making the function continuous at that point.

Continuity at a Point

A function is continuous at a point if the limit of the function as x approaches that point equals the function's value at the point. This definition ensures there is no break, jump, or hole in the function at that location.

Limit of a Function

The limit of a function as x approaches a point is the value that the function approaches, regardless of the function's actual value at that point. It forms the basis for understanding the behavior of functions as inputs get arbitrarily close to a given point.

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