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If lim_X-1 f(x) - 2/X-1=3, find lim_X-1 f(x).

Answer :

Final answer:

To find the limit lim_x->1 f(x), substitute x = 1 into f(x) - 2/(x-1) and find the value of f(1), which is 5.

Explanation:

To find the limit limx->1 f(x), we can use the given information about the limit of the expression f(x) - 2/(x-1). Since the limit of the expression is 3, we can substitute x = 1 into the expression to find the value of f(1). Therefore, the limit of f(x) as x approaches 1 is the same as the value of f(1), which we found to be 5.

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Rewritten by : Barada

Final Answer:

The limit of f(x) as x approaches 1 is 5.

Explanation:

To find the limit of f(x) as x approaches 1, we can use the limit properties and algebraic manipulation. Given that:

[tex]\lim_{x \to \ 1} [f(x) - 2/(x - 1)] = 3[/tex]

We can rearrange the equation as follows:

lim(x -> 1) f(x) - lim(x -> 1) 2/(x - 1) = 3

Now, let's find the limit of the second term on the left side:

lim(x -> 1) 2/(x - 1) = 2/0

This is an indeterminate form (2/0), indicating that further simplification is needed. We can rewrite 2/(x - 1) as follows:

2/(x - 1) = 2/[(x - 1) * (1/x)]

Now, we can see that as x approaches 1, both (x - 1) and (1/x) approach 0. Therefore, we can apply L'Hôpital's Rule:

lim(x -> 1) 2/[(x - 1) * (1/x)] = lim(x -> 1) [2'/(x)] = 2/1 = 2

So, we have:

lim(x -> 1) f(x) - 2 = 3

Now, solving for the limit of f(x):

lim(x -> 1) f(x) = 3 + 2 = 5

Therefore, the limit of f(x) as x approaches 1 is 5.

Limits and L'Hôpital's Rule can be powerful tools for solving complex mathematical problems involving functions and indeterminate forms like 2/0. Understanding these concepts is essential in calculus and mathematical analysis.

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