<s>[INST] Suppose $\lim _{ x\to 0 } \frac { a-\sqrt { { a }^{ 2 }-{ x }^{ 2 } }-{ x }^{ 2 }/4 }{ { x }^{ 4 } } $ is finite then how to find the value of a? Suppose $\lim _{ x\to 0 } \frac { a-\sqrt { { a }^{ 2 }-{ x }^{ 2 } }-{ x }^{ 2 }/4  }{ { x }^{ 4 } } $ is finite then how to find the value of a ?
I'm having trouble understand the fact that even though the highest power of x in the denominator and numerator is not same,how can the limit be finite?
Yeah I know,probably i'm having a conceptual doubt.Can someone help please!
 [/INST] HINT:
$$a-\sqrt{a^2-x^2}-\dfrac{x^2}4$$
$$=\dfrac{a^2-(a^2-x^2)-\dfrac{x^2(a+\sqrt{a^2-x^2})}4}{a+\sqrt{a^2-x^2}}$$
$$=x^2\cdot\dfrac{4-a-ax^2-\sqrt{a^2-x^2}}{4(a+\sqrt{a^2-x^2})}$$
$$=x^2\cdot\dfrac{(4-a-ax^2)^2-(a^2-x^2)}{4(a+\sqrt{a^2-x^2})(4-a+\sqrt{a^2-x^2})}$$
$$=x^2\cdot\dfrac{16-8a+x^2\{1-2a(4-a)\}+a^2x^4}{4(a+\sqrt{a^2-x^2})(4-a+\sqrt{a^2-x^2})}$$
So, we need $16-8a=0$
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