Math

QuestionDoes the function f(x)=xf(x)=x have a limit as xx approaches 3 from all real numbers except 3?

Studdy Solution

STEP 1

Assumptions1. The function is f(x)=xf(x)=x . The domain of the function is all real numbers except for x=3x=3

STEP 2

The limit of a function at a certain point is the value that the function approaches as xx approaches that point. We need to check if the function f(x)=xf(x)=x has a limit at all points in its domain.

STEP 3

For all real numbers except x=3x=3, the function f(x)=xf(x)=x is continuous. A function is continuous at a point if the limit exists at that point and is equal to the function's value at that point. Therefore, the limit of f(x)=xf(x)=x exists for all real numbers except x=3x=3.

STEP 4

To find the limit of the function at x=3x=3, we need to check the limit from both the left side (as xx approaches 33 from values less than 33) and the right side (as xx approaches 33 from values greater than 33).

STEP 5

Calculate the limit as xx approaches 33 from the left side.
limx3f(x)=limx3x\lim{{x \to3^-}} f(x) = \lim{{x \to3^-}} x

STEP 6

Since f(x)=xf(x)=x for all xx in the domain, the limit as xx approaches 33 from the left side is 33.
limx3f(x)=3\lim{{x \to3^-}} f(x) =3

STEP 7

Calculate the limit as xx approaches 33 from the right side.
limx3+f(x)=limx3+x\lim{{x \to3^+}} f(x) = \lim{{x \to3^+}} x

STEP 8

Since f(x)=xf(x)=x for all xx in the domain, the limit as xx approaches 33 from the right side is also 33.
limx3+f(x)=3\lim{{x \to3^+}} f(x) =3

STEP 9

Since the limit from the left side is equal to the limit from the right side, the limit of the function f(x)=xf(x)=x at x=3x=3 exists and is equal to 33.
Therefore, the function f(x)=xf(x)=x has a limit at all points in its domain, which is all real numbers except for x=3x=3.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord