limit of a function being a sum and product of trignometric and hyperbolic functions

Question

I need to know the $$\lim_{(x,y)\to(0,0)}{\sin x-\sinh x\over 2x^2+y^2-xy}\ln(x^2+y^2)$$I used finite expansion on the hyperbolic and trigonometric function and then used polar coordinates where the limit tends then to $0.$

Am I right ?

Answer 1

If $x=0,y\ne 0,$ the expression equals $0$. This tells us the limit, if it exists, is $0.$ For $x\ne 0,$ use the McClaurin series for $\sin x, \sinh x$ to see the numerator is $O(x^3).$ The denominator equals $7x^2/4 +(y-x/2)^2.$ So for $(x,y)$ close to $(0,0),x\ne 0,$ the expression equals$$\frac{|O(x^3)|}{7x^2/4 +(y-x/2)^2}|\ln(x^2+y^2)|\le \frac{|O(x^3)|}{7x^2/4}|\ln(x^2)|.$$

Since $\lim _{x\to 0}|x\ln x^2|=0,$ we see the limit in question is $0.$