Is this correct? Does no limit exist?

Again we apply L'Hospitals Rule:

$$\frac{1}{2}\cdot\lim_{x\to0} \frac{8x\cdot\cos(4x^2)}{\color{red}2x} = \frac{1}{2}\cdot\lim_{x\to0}\color{red}4\cdot\cos(4x^2) =\color{red}2 \lim_{x\to0} \cos(4x^2) = \color{red}2 $$


I think the best approach is to just make the substitution $u = x^2 + y^2$.

Then $g$ is continuous at $(0,0)$ if and only if $$\lim_{u\to0^{+}}\frac{\sin(2u)}{u}=a$$ but then, since $$\lim_{u\to0^{+}}\frac{\sin(2u)}{u}=2\left(\lim_{u\to0^{+}}\frac{\sin(2u)}{2u}\right)=2(1)=2$$ we get continuity at $(0,0)$ if and only if $a = 2$.

Tags:

Limits

Continuity

Multivariable Calculus

Related

Proving $\left(1-\frac13+\frac15-\frac17+\cdots\right)^2=\frac38\left(\frac1{1^2}+\frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\cdots\right)$ Expected number of draws to find a match How to find the Riemannian Distance on the Sphere $S^n\subset \Bbb{R}^{n+1}$ Proof that the square of an odd number is of the form $8m+1$ for some integer $m$ Why does factorizing of polynomials work Can we prove a stronger claim? Linear Algebra Proof of Mean Value Theorem Determine sinus and cosinus : $2\sin x + 3\cos x = 3$ If $\alpha,a,b$ are integers and $b\neq-1$, then prove that, if $\alpha$ satisfies the equation $x^2+ax+b+1=0$, $a^2+b^2$ must be composite. A conjecture regarding the eigenvalues of real symmetric matrices Solving the differential equation $q \vec{v} \times \vec{B} + q \vec{E} = m \frac{d\vec{v}}{dt}$ What makes the Cauchy principal value the "correct" value for a integral?

Recent Posts