Q. 92

Question

Prove the difference rule for limits by applying the sum and constant multiple rules for limits.

Step-by-Step Solution

Verified

Answer

$\lim_{x \to c} (f (x) - g (x)) = \lim_{x \to c} f (x) - \lim_{x \to c} g (x)$

1Step 1. Given Information:

The strategy is to prove the difference rule of limit applying the sum and constant multiple rules for limit.

2Step 2. Prove:

The difference rule for limit is given by:

$\underset{x \to c}{l i m} (f (x) - g (x)) = \underset{x \to c}{l i m} f (x) - \underset{x \to c}{l i m} g (x)$

Now take the left-hand side we get,

$\underset{x \to c}{l i m} (f (x) - g (x)) = \underset{x \to \infty}{l i m} [f (x) + (- 1) g (x)] = \underset{x \to c}{l i m} f (x) + \underset{x \to c}{l i m} (- 1) g (x) [S u m r u l e] = \underset{x \to c}{l i m} f (x) + (- 1) \underset{x \to c}{l i m} g (x) [C o n s t a n t m u l t i p l e r u l e] = \underset{x \to c}{l i m} f (x) - \underset{x \to c}{l i m} g (x)$

Hence, the difference rule can be proved by applying the sum and constant multiple rules.

Q. 91

Q. 94

Other exercises in this chapter

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Q. 91

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Q. 94

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Q. 95

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