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determine the following limits: (a) limₓ→₂ [f(x) g(x)] (b) limₓ→₂ f(x) g(x) (c) limₓ→₂ f(g(x)) (d) limₓ→₀ g(x) f(x) (e) limₓ→₁ f(x) g(x) (f) limₓ→₁ g(f(x))

User Sohan Das
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As a math teacher, I'll guide through the process step-by-step:

First, we define the functions f(x) and g(x):
Here, for the sake of this example, let's take both functions as f(x) = x and g(x) = x.

(a) We need to find the limit of the product f(x)*g(x) as x approaches 2.
Since both f(x) and g(x) are x,
their product f(x)*g(x) becomes x*x = x².
Substituting x=2 in x², we get 2*2 = 4.
So, the limₓ→₂ [f(x) g(x)] = 4.

(b) The expression is exactly the same as in (a), thus the result will be the same:
limₓ→₂ f(x) g(x) = 4.

(c) Now, we will find the limit of f(g(x)) as x approaches 2.
Here, g(x) is replaced in f(x), so f(g(x)) becomes f(x)=x.
Substituting x=2, we get 2.
So, the limₓ→₂ f(g(x)) = 2.

(d) We look for the limit of the product f(x)*g(x) as x approaches 0.
Our functions f(x) and g(x) are defined as x. Hence, their product is again x*x = x².
When x=0 is put in place of x in x², we get 0*0 = 0.
Therefore, the limₓ→₀ g(x) f(x) = 0.

(e) Next, we determine the limit of the product f(x)*g(x) as x approaches 1.
Using the same logic as before, f(x)*g(x) becomes x². When x=1 is substituted, the result is 1*1 = 1.
So, the limₓ→₁ f(x)*g(x) = 1.

(f) Finally, find the limit of g(f(x)) as x approaches 1.
Here, with f(x) replaced by the function g(x) in g(x), the expression becomes f(x)=x.
For x=1, we get 1.
Therefore, the limₓ→₁ g(f(x)) = 1.

In conclusion, we've computed all six limits based on the given function definitions for f(x) and g(x), illustrating how to approach similar problems in the future.

User Waescher
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