Find the limits. f(x)=4-x^2, g(x)=√(x+1) (a) limx →1 f(x) (b) limx →3 g(x) (c) limx →1 g(f(x))

Name: Problem 25, Chapter 1: Calculus
Uploaded: 2021-09-21T16:29:56-08:00
Duration: 1 min 39 s
Channel: Nick Johnson
Description: Find the limits. f(x)=4-x^2, g(x)=√(x+1) (a) limx →1 f(x) (b) limx →3 g(x) (c) limx →1 g(f(x))

step 1 Hello. So here we have f of x equal to 4 minus x squared, and g of x is equal to the square root

Find the limits. f(x)=4-x^2, g(x)=√(x+1) (a) limx →1 f(x)...

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Key Concepts

Limit of a Function

A limit of a function describes the value that the function approaches as the input gets arbitrarily close to a particular point. Computing limits is fundamental in calculus and is used to understand the behavior of functions near points of interest, even if the function is not defined exactly at those points.

Continuity

Continuity is a property of functions where small changes in the input lead to small changes in the output, meaning the function has no sudden jumps or breaks. For continuous functions, the limit as x approaches a point is simply the value of the function at that point, allowing for limits to be evaluated by direct substitution.

Composition of Functions in Limits

When dealing with the limit of a composite function, if the inner function has a limit as the variable approaches a certain value and the outer function is continuous at that limit, then the limit of the composite function can be computed by applying the outer function to the limit of the inner function. This concept simplifies the evaluation of complex limits by breaking them down into simpler parts.

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