Question

Limit x tends to a (x-a) ^x-a

Answer 1

Answer:

Step-by-step explanation:

If we directly put the limit it will be 0^0 which is a intermediate form so we can change it to 0/0 or infinity/infinity form then apply L hospitals rule by differentiating numerator and demonater to simple form and put limits and get the required answer

Lim_x->a (x-a)^(x-a)

Here f(x)=x-a

g(x)=x-a

Change it to e^[limx->a g(x)/(1÷(ln(f(x)))]

Try yourself and find answer

Ty

Answer 2

The value of limit x tends to a (x-a) ^x-a is 1.

Given: Limit x tends to a (x-a) ^x-a

To Find: The value of limit x tends to a (x-a) ^x-a

Solution:

• We need to achieve a form of 0/0 or ∞/∞ to apply L'Hospital rule.
• In L'Hospital rule, we differentiate the numerator and denominator separately to find a finite value of the limit.

The expression given is

         Let y = Limit x tends to a (x-a) ^x-a

We can see, that on putting x = a in the expression, we get, the 0^0 form.

So we take to log on both sides,

        log y = Limit x → a ( x - a ) log ( x - a )

    ⇒ log y = Limit x → a ( log ( x - a ) / ( 1 / ( x - a ) ))  [ ∞/∞ form ]

As we can see, we have reached, ∞/∞ form, so we can apply  L'Hospital rule, where we shall differentiate the numerator and denominator separately to find a finite value of the limit.

    ⇒ log y = Limit x → a (1 / ( x - a )) ÷ ( 1 / ( x - a )² )

    ⇒ log y = Limit x → a  ( x - a )

Putting x = a, we get,

    ⇒ log y = 0

Taking exponentiation on both sides we get,

    ⇒ e ^ log y = e^0                           [ As e^log a = a ]

    ⇒   y = 1      

Hence, the value of limit x tends to a (x-a) ^x-a is 1.

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