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Lecture 5

MATA31H3 Lecture 5: W5 Complete Lectures

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Natalie Rose

MATT A 31 Week 5: Complete Lectures Textbook: Taalman, Kohn: US. Single Variable. 5.1 Basic Limits & Limit Laws. 2 Le Numerical investigations s limits and typically suggest the correct value of a limit. uld fi In the Week 4 Lectures f general functi The p To prove that lim(ax b)- ac b we first establish the connection between and o b/ hen 0 lx have: Kax +b QED g But most limit computations are based neither on merely suggestive numerical estimates nor on direct (precise) application of the definition of limit. Instead, such computations are performed most easily and based tually h th ally definition of limit (o,E proofs). We'll prove several theorems and leave the proofs ofthe remaining theorems as homework problems. Knowing the Uniqueness of Limit theorem (below is the alternative proof of the theorem based on algebraic s of combinations of functions. defi THEOREM Uniqueness of Limit: If limfox) L and limf(x)- M,then L M. PROOF that ha f the th that the trary to th Suppo L, MM, the th h that if 0 h 0 the L-M L -M have f(x)-L Ch g Eu the th th equal to M L-M L-M have f Ch Triangle inequal L-fi or L-MI IL-MIwhich contradicts the supposition that function can have more than one limit. Therefore que. QED proved that th having kn rithm of functions. limits, we can use several easy rules (Limit Laws). ith L,M,c, k ER Constant La K-K is equal to Kmeans that for each 0 th exists h that if Proof: L K 0 K-K no matter ho KKE, Since K- 2. Addition Law Constant multiple Law limlk f(r kl. 4. Reciprocal Law lim We need to prove that there exists such 0that if 00 and E 0 that if x c is in the interv al (L c +oi) then g( h >0 that if o 0th ally. lim g(x) Lif fo If L ,then the exist such >0 and E 0 that if x c is in the intcrval (c-o.c+o,) then h is in the interval (L or L-E0 that if 0 lx-cl then E h (y) B-P C-a, But values of f(r) on (c-p,c p) x c are between val of g(x) and h(x) the whenever o K-cl
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