In Exercises 37-54, find the limit (if it exists). limx →4^- (x^2)/(x^2+16)

Name: Problem 42, Chapter 1: Calculus of a Single Variable
Uploaded: 2020-11-03T03:58:03-08:00
Duration: 3 min 2 s
Channel: Samuel Goyette
Description: In Exercises 37-54, find the limit (if it exists). limx →4^- (x^2)/(x^2+16)

step 1 Okay, so in this problem, we look at this limit, x squared divided by x squared plus 16, when x

In Exercises 37-54, find the limit (if it exists). limx →4^-...

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Key Concepts

Continuity and Direct Substitution

If a function is continuous at a point (meaning there is no abrupt jump, hole, or asymptote), the limit as the function approaches that point can be found by directly substituting the value into the function. This simplifies the process of evaluating limits, as no algebraic manipulation or limit laws are necessary for functions that are continuous at the point of interest.

One-Sided Limit

A one-sided limit considers the value that the function approaches as the input variable approaches a specific point from one side—either from the left or the right. In these cases, the analysis only involves values on one side of the point, which is important for handling functions that may behave differently on each side.

Limit

In calculus, the limit of a function describes the value that the function approaches as the input variable gets arbitrarily close to a particular point. This concept is fundamental for understanding continuous behavior and for defining derivatives and integrals.

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