plz help me with this,when f(x)=1/(x^2+2), find lim[f(x)-f(a)]/x-a
x ---->a
To find the limit of the expression [f(x) - f(a)] / (x - a) as x approaches a, we can use algebraic manipulation and the definition of the derivative.
First, let's calculate f(x):
f(x) = 1 / (x^2 + 2)
Next, let's calculate f(a):
f(a) = 1 / (a^2 + 2)
Now, we can substitute these values into the expression:
[f(x) - f(a)] / (x - a) = [1 / (x^2 + 2) - 1 / (a^2 + 2)] / (x - a)
To simplify this expression, we need to find a common denominator. The common denominator for (x^2 + 2) and (a^2 + 2) is (x^2 + 2)(a^2 + 2).
[f(x) - f(a)] / (x - a) = [(a^2 + 2) - (x^2 + 2)] / [(x^2 + 2)(a^2 + 2)] / (x - a)
= (a^2 + 2 - x^2 - 2) / (x - a)(x^2 + 2)(a^2 + 2)
= (a^2 - x^2) / (x - a)(x^2 + 2)(a^2 + 2)
Now, we can simplify further by factoring (a^2 - x^2) as (a - x)(a + x):
[f(x) - f(a)] / (x - a) = [(a - x)(a + x)] / (x - a)(x^2 + 2)(a^2 + 2)
Finally, we can cancel out the common factor of (x - a) in the numerator and denominator:
[f(x) - f(a)] / (x - a) = (a + x) / (x^2 + 2)(a^2 + 2)
Now, we can take the limit as x approaches a:
lim [f(x) - f(a)] / (x - a) = lim (a + x) / (x^2 + 2)(a^2 + 2)
= (a + a) / (a^2 + 2)(a^2 + 2)
= 2a / (a^2 + 2)(a^2 + 2)
Therefore, the limit of [f(x) - f(a)] / (x - a) as x approaches a is 2a / (a^2 + 2)(a^2 + 2).
Note: When finding limits, it is important to simplify the expression as much as possible and use algebraic manipulation techniques to avoid division by zero.