Question

if lim x tends to 2 (x^p-2^p/x-2)=192. the p= (3,4,6,10)​

Answer 1

SOLUTION

GIVEN

$\displaystyle \sf{\lim_{x \to 2} \: \frac{ {x}^{p} - {2}^{p} }{x-2} = 192}$

TO DETERMINE

The value of p

EVALUATION

Here the given limit is

$\displaystyle \sf{\lim_{x \to 2} \: \frac{ {x}^{p} - {2}^{p} }{x-2} = 192}$

We are aware of the formula that

$\boxed{ \: \: \displaystyle \sf{\lim_{x \to a} \: \frac{ {x}^{n} - {a}^{n} }{x-a} = n \: {a}^{n - 1} } \: \: }$

Thus we get

$\displaystyle \sf{\lim_{x \to 2} \: \frac{ {x}^{p} - {2}^{p} }{x-2} = 192}$

$\displaystyle \sf{ \implies \: p \: \times {2}^{p - 1} = 192}$

$\displaystyle \sf{ \implies \: p \: \times {2}^{p - 1} = 6 \times {2}^{5} }$

$\displaystyle \sf{ \implies \: p \: \times {2}^{p - 1} = 6 \times {2}^{6 - 1} }$

Comparing both sides we get

p = 6

FINAL ANSWER

Hence the required value of p = 6

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