Question

State whether the following statements are true or false. Justify your answer.

a). \lim_{x \to 0 } \frac{1}{x^2} - \frac{1}{sin^2 x} is in \frac{0}{0} form

b). f(x,y) = sin\tfrac{x^2 y}{x^2 + y^2}  / ln\tfrac{x+y}{x}

  Image ignouacademy-com--p-whether-49793 is a homogeneous function of degree 2.

c). Domain of f(x,y) =\frac{xy}{x^4+ y^4} is R2

Image ignouacademy-com--p-false-70465

d). The function f(x,y) = (x3 y + 1,x2 + y2) is locally invertible at (1,2).

e). The function f (x, y ) = x3 + y3   is integrable on [1,2] - [1,3].

 

 

 

12 Feb 2018
Answer :
Word Count : 475
### Problem a) Statement: \(\lim_{x \to 0} \left( \frac{1}{x^2} - \frac{1}{\sin^2 x} \right)\) is in \(\frac{0}{0}\) form. Solution: 1. Evaluate the limit of each term individually: - \(\lim_{x \to 0} \frac{1}{x^2} = +\infty\) - \(\lim_{x \to 0} \frac{1}{\sin^2 x} = +\infty\) 2. Subtract the two limits: - \(\infty - \infty\) is an indeterminate form, not \(\frac{0}{0}\). The statement is false. The limit is of the form \(\infty - \infty\), not \(\frac{0}{0}\). ### Problem b) Statement: \(f(x, y) = \frac{\sin\left(\frac{x^2 y}{x^2 + y^2}\right)}{\ln\left(\frac{x + y}{x}\right)}\) is a homogeneous function of degree 2. Solution: 1. Definition of a homogeneous function: A function \(f(x, y)\) is homogeneous of degree \(k\) if \(f(tx, ty) = t^k f(x, y)\) for all \(t > 0\). 2. Check homogeneity: - Substitute \(tx\) and \(ty\) into
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  Image ignouacademy-com--p-following-34338 is a homogeneous function of degree 2.

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