Limit Detective: Product and Quotient Rules

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A limit puzzle using product and quotient rules.

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Keith benedict Avila

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Limit Detective: Product and Quotient RulesVersión en línea A limit puzzle using product and quotient rules. por Keith benedict Avila 1 As x approaches 1, I’m the limit of (x^2 - 1) divided by (x - 1). What am I? Hints 1 Recognize the remaining expression is x+1, so at x = 1 it equals 2. 2 Cancel the common factor (x-1) and then substitute x = 1. 3 Factor the numerator as a difference of squares: (x-1)(x+1). 2 I handle the way two changing things multiply as they approach a point. What rule tells us that lim x->a f(x)g(x) = f(a)g(a) when both f and g settle nicely at a? Hints 1 The result is the product of the two limits. 2 Requires the limits of both functions at the point to exist. 3 Involves multiplying two functions. 3 When a single thing is on top and another on bottom, their changing rates combine to form a new rate. Which rule gives lim x->a f(x)/g(x) via (f'g - fg')/g^2? Hints 1 G(a) must not be zero at the limit point. 2 Uses derivatives of numerator and denominator. 3 It's about division of functions. 4 Consider applying limit laws to simple expressions: sums, products, and quotients of limits. Which broad set of rules helps you evaluate these without redoing the entire process? Hints 1 Knowing continuity or existance of limits at a is helpful. 2 They let you break complex expressions into simpler parts. 3 These laws cover combining limits of f and g.

Limit Detective: Product and Quotient RulesVersión en línea

A limit puzzle using product and quotient rules.

por Keith benedict Avila

As x approaches 1, I’m the limit of (x^2 - 1) divided by (x - 1). What am I?

Hints

Recognize the remaining expression is x+1, so at x = 1 it equals 2.

Cancel the common factor (x-1) and then substitute x = 1.

Factor the numerator as a difference of squares: (x-1)(x+1).

I handle the way two changing things multiply as they approach a point. What rule tells us that lim x->a f(x)g(x) = f(a)g(a) when both f and g settle nicely at a?

Hints

The result is the product of the two limits.

Requires the limits of both functions at the point to exist.

Involves multiplying two functions.

When a single thing is on top and another on bottom, their changing rates combine to form a new rate. Which rule gives lim x->a f(x)/g(x) via (f'g - fg')/g^2?

Hints

G(a) must not be zero at the limit point.

Uses derivatives of numerator and denominator.

It's about division of functions.

Consider applying limit laws to simple expressions: sums, products, and quotients of limits. Which broad set of rules helps you evaluate these without redoing the entire process?

Hints

Knowing continuity or existance of limits at a is helpful.

They let you break complex expressions into simpler parts.

These laws cover combining limits of f and g.

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Limit Detective: Product and Quotient Rules

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Limit Detective: Product and Quotient RulesVersión en línea

por Keith benedict Avila

As x approaches 1, I’m the limit of (x^2 - 1) divided by (x - 1). What am I?

Hints

I handle the way two changing things multiply as they approach a point. What rule tells us that lim x->a f(x)g(x) = f(a)g(a) when both f and g settle nicely at a?

Hints

When a single thing is on top and another on bottom, their changing rates combine to form a new rate. Which rule gives lim x->a f(x)/g(x) via (f'g - fg')/g^2?

Hints

Consider applying limit laws to simple expressions: sums, products, and quotients of limits. Which broad set of rules helps you evaluate these without redoing the entire process?

Hints

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