Multivariable Limits

Definition

• With multiple variables, the direction you approach a function can change the value it approaches, so we need for ${\lim }_{P\to P_0}f(P)=L$
  1. Values of $f(x)$ get arbitrarily close to $L$
  2. Values of $f(x)$ stay close to $L$ for all $P$ sufficiently close to $P_0$
• We can notate that as
  1. $\text{Fix }ϵ>|f(p)-L|\ge 0$
     • Epsilon is an arbitrary number
  2. There exists $\delta >0$ such that (1) holds whenever $|P{P}_0|>\delta$

Pure math notation:

$$\text{Fix }ϵ>0\exists \delta >0|f(P)-L|<ϵ\text{ whenever }0<|P{P}_0|<\delta$$

Continuity

• Theorem
  • If
    1. $g(t)$ is continuous
    2. $t(x,y,\dots )$
  • Then $f(x,y,\dots )$ is continuous
• Means we don’t need to do all that definition work when functions are made of continuous functions such as
  • Polynomials
  • Rational (where the denominator $\ne 0$
  • Composition of elementary functions ($e^t$, $\sin t$, etc.)

Solving

1. If continuous, just plug in
2. Substitution Rule
   • Can sometimes turn multidimensional limit into 1D limit
   • ${\lim }_{P\to P_0}f(P)={\lim }_{t\to t_0}f(t)=L$
   • Conditions
     • $f(x,y,\dots )=g(t)$
     • $t(x,y,\dots )$ is continuous at $(x_0,y_0,\dots )⟺t(x_0,y_0,\dots )=t_0$
     • ${\lim }_{t\to t_0}g(t)=L$
3. Limit Along Curves
   1. Predict the value
      • Plug a curve with variable slope into the limit
      • Possible functions
        • $x=t,y=at$
        • $x=t,y=a{t}^2$
        • $x=t\cos \theta ,y=t\sin \theta$
      • ${\lim }_{t\to 0}f(t,at,\dots )$
        • DNE → multi-var limit DNE
        • L(a) exists but depends on the slope → multi-var limit DNE
        • L exists but independent of the slope → verify that value of L
      • If you try multiple curves and they produce different values, the limit DNE
   2. Verify L
      • Using Squeeze Theorem
      • Conditions
        • $|f(P)-L|\le h(R)$
        • ${\lim }_{R\to 0}h(R)=0$
4. Squeeze Theorem
   • Start with the definition and then keep simplifying
     1. $|f(x)-L|$
     2. first $\le$ replace $x$ and $y$ with $R$
     3. keep chaining $\le$s to simplify further
     4. if the limit of what you get is 0, then the limit of the original is 0
   • Good inequalities to use
     • $|\sin x|\le |x|$
     • $\frac{A}{B}\le \frac{A}{C}\text{ if }C\le B$
     • $A^n\le A^{n-1}\text{ if }A\le 1$

$$R=\sqrt{x^2+y^2},|x|\le R,|y|\le R$$