Property of limit

Calculus Level 1

Two functions f ( x ) f(x) and g ( x ) g(x) satisfy the following equations: lim ⁡ x → a f ( x ) x − a = 3 , lim ⁡ x → a g ( x ) x − a = 2. \lim_{x\rightarrow a}\frac{f(x)}{x-a}=3, \ \lim_{x\rightarrow a}\frac{g(x)}{x-a}=2. Find lim ⁡ x → a 2 f ( x ) + 3 g ( x ) f ( x ) − g ( x ) . \lim_{x \rightarrow a} \frac{2f(x)+3g(x)}{f(x)-g(x)}.


The answer is 12.

Sandeep Mehra by Sandeep Mehra , 23, India

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1 solution

As we know, limit of functions in an arithmetic orientation can work out a arithmetic resultant of limits of every functions in it. Seems hard? Consider this:

lim ⁡ x → a ( f ( x ) − g ( x ) ) = lim ⁡ x → a f ( x ) − lim ⁡ x → a g ( x ) \lim_{x \to a}(\mathcal{f}(x)-\mathcal{g}(x))=\lim_{x \to a}\mathcal{f}(x)-\lim_{x \to a}\mathcal{g}(x)

Another property: lim ⁡ x → a c f ( x ) = c lim ⁡ x → a f ( x ) \lim_{x \to a} c \mathcal{f}(x) = c \lim_{x \to a} \mathcal{f}(x)

Here, in this case, let's divide each f ( x ) \mathcal{f}(x) & g ( x ) \mathcal{g}(x) by ( x − a ) (x-a) , which doesn't change anything by sure.

Then, using the properties I've shown, we can simply say, the answer is: 2 × 3 + 3 × 2 3 − 2 = 12 \frac{2 \times 3 + 3 \times 2}{3-2}=12

0 Helpful 0 Interesting 0 Brilliant 0 Confused

I have a feeling that this answer is wrong... because you're taking x-a as a constant when you shouldn't... because you're taking the limit from x approaching a so... dividing by x-a is like dividing by 0 hence exploding the limit to infinity. I think it's an indeterminate limit that way... I don't know. I just don't think it's right.

stefanos charkoutsis - 2 years, 9 months ago

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