Proof
(related to Proposition: Arithmetic of Functions with Limits - Sums)
Context
Hypothesis
- Let f,g have the limits \lim_{x\to a}f(x)=L and \lim_{x\to a}g(x)=H.
- Given \epsilon > 0.
Implications
- By the definition of limit:
- there is a \delta_1 > 0 such that if x\in D and 0 < |x-a| < \delta_1, then |f(x)-L|\le \frac \epsilon2, and
- there is a \delta_2 > 0 such that if x\in D and 0 < |x-a| < \delta_2, then |g(x)-H|\le \frac \epsilon2.
- Get the minimum \delta=\min(\delta_1,\delta_2).
- Note that \delta > 0.
- Since f+g:D\to\mathbb R, we have that if x\in D with 0 < |x-a| < \delta, then by the triangle inequality |(f(x)+g(x))-(L+H)|=|(f(x)-L)+(g(x)-H)|\le |f(x)-L|+|g(x)-H| < \frac \epsilon2 + \frac \epsilon2=\epsilon.
Conclusion
- By the definition of limit, this shows that the limit of the sum of both functions is given by \lim_{x\to a}(f(x)+g(x))=L+H.
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References
Bibliography
- Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016