Evaluate the limit. limx →4[(1)/(√(x)-2)-(4)/(x-4)] |...

Name: Problem 18, Chapter 4: Calculus for AP
Uploaded: 2019-07-11T13:57:26-08:00
Duration: 2 min 27 s
Channel: Bahar Tehranipoor
Description: evaluate the limit. limx →4[(1)/(√(x)-2)-(4)/(x-4)]

Key Concepts

Limits

Limits describe the behavior of functions as the independent variable approaches a specific value. They are fundamental in calculus for understanding continuity, derivatives, and integrals, and often require careful analysis when direct substitution leads to undefined expressions.

Indeterminate Forms

Indeterminate forms occur when substitution into a limit results in ambiguous expressions like 0/0 or ? - ?. Identifying these forms is crucial because they indicate that further algebraic manipulation or alternative methods are necessary to evaluate the limit correctly.

Algebraic Simplification

Algebraic simplification, such as combining fractions, factoring, or rationalizing, is a key strategy in resolving limits that initially present indeterminate forms. This process often involves rewriting the expression to cancel out problematic terms, thereby allowing for direct evaluation of the limit.

Rationalization and Factorization

Rationalization involves multiplying by a conjugate to eliminate radicals, while factorization entails expressing expressions as products of simpler factors. Both techniques are instrumental in simplifying expressions that involve square roots or differences in powers, which commonly arise in limit problems.

Transcript

00:01 In this exercise, we want to evaluate the limit.

00:03 So what we can do is first see if we can use lopitow's rule.

00:06 So we would do that by seeing if we get an indeterminate form when we have x approach 4 for what we have here.

00:12 So if we have x approach 4 in 1 over the square root of x minus 2, we're going to get infinity because we're just going to have a denominator that is going to be really, really close to zero, but the numerator is going to be the same.

00:23 So the number is just going to keep getting bigger and bigger, so approaching infinity.

00:26 Then we have this minus sign right here.

00:28 Then we have 4 over x minus 4.

00:30 So if we have x approach 4 here, we get the same thing.

00:32 We have the numerators stay the same, but the denominator get closer and closer to zero.

00:36 So the number is going to get really large.

00:37 So we'll have infinity minus infinity, which is in fact indeterminate.

00:41 So we can use lopi tells.

00:43 So what we're going to do is we're going to first combine this into a single fraction.

00:48 And one thing that's really important to note is that if we have the square root of x plus 2 times the square root of x minus 2, they get x minus 4.

00:57 So what we can do to get x minus 4 in this denominator is multiplied by the square root of x plus 2 over the square root of x plus 2.

01:04 So we get the square root of x plus 2 over x minus 4 minus 4, which is what we're trying to limit of...

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