3.) Compute the limits: g limit (x,y) → (0₁1) By ܘ
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Transcribed Image Text:3. Compute the limits:
limit
(x,y) → (0,1)
ها
+³y
(d
7
X-D
poa tamr
![Theorem 4 Properties of Continuous Functions Suppose that f: AC
R"→ Rm, g: ACR" → Rm, and let c be a real number.
(i) If f is continuous at xo, so is cf, where (cf)(x) = c[ƒ(x)].
(ii) If f and g are continuous at xo, so is f+g, where the sum of f and g is
defined by (f+g)(x) = f(x) + g(x).
(iii) If f and g are continuous at x and m = 1, then the product function fg
defined by (fg)(x) = f(x)g(x) is continuous at Xo.
(iv) If f: AC R"→ R is continuous at xo and nowhere zero on A, then the
quotient 1/f is continuous at xo, where (1/f)(x) = 1/f(x).
(v) If f: A CR" → R" and f(x) = (fi(x), ..., fm(x)), then f is continuous
at xo if and only if each of the real-valued functions fi, ..., fm is continuous
at Xo.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa03415f8-38f2-4478-b688-afaca619ca5c%2F447dd3d9-d29b-4ff6-a133-a1c197e9ee9d%2Fum9xy8g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Theorem 4 Properties of Continuous Functions Suppose that f: AC
R"→ Rm, g: ACR" → Rm, and let c be a real number.
(i) If f is continuous at xo, so is cf, where (cf)(x) = c[ƒ(x)].
(ii) If f and g are continuous at xo, so is f+g, where the sum of f and g is
defined by (f+g)(x) = f(x) + g(x).
(iii) If f and g are continuous at x and m = 1, then the product function fg
defined by (fg)(x) = f(x)g(x) is continuous at Xo.
(iv) If f: AC R"→ R is continuous at xo and nowhere zero on A, then the
quotient 1/f is continuous at xo, where (1/f)(x) = 1/f(x).
(v) If f: A CR" → R" and f(x) = (fi(x), ..., fm(x)), then f is continuous
at xo if and only if each of the real-valued functions fi, ..., fm is continuous
at Xo.
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