Show that the function f(x) = \(\begin{cases} (1 + 3x)^{\frac{1}{x}}

1.	Show that the function f(x) = \(\begin{cases} (1 + 3x)^{\frac{1}{x}} & x \neq 0 \\e^3 &x = 0 \end{cases}\) is continuous at x = 0
Answer» \(\lim\limits_{x \to 0}(1 + 3x)^{\frac{1}{3x}\times 3}\) = e³ Also f(x) at x = 0 is e³ i.e f(0) = e³ ∴ \(\lim\limits_{x \to 0}\) f(x) = f(0) = e³ ∴ f(x) is continuous at x = 0

Show that the function f(x) = \(\begin{cases} (1 + 3x)^{\frac{1}{x}} & x \neq 0 \\e^3 &x = 0 \end{cases}\) is continuous at x = 0

Answer»

\(\lim\limits_{x \to 0}(1 + 3x)^{\frac{1}{3x}\times 3}\) = e³

Also f(x) at x = 0 is e³

i.e f(0) = e³

∴ \(\lim\limits_{x \to 0}\) f(x) = f(0) = e³

∴ f(x) is continuous at x = 0

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