(a) Let f: ℝ→ℝ be given by f(x) = e^-x^2. Does the limit limx → 0 f(x) exist? If so, what is it.

Step 1: For part (a), we are given the function f(x) = e^(-x^2). To find the limit lim(x->0) f(x

Question

(a) Let $f: \mathbb{R} \to \mathbb{R}$ be given by $f(x) = e^{-x^2}$. Does the limit $\lim_{x \to 0} f(x)$ exist? If so, what is it. (b) Let $g: \mathbb{R}^2 \to \mathbb{R}$ be given by $g(x,y) = e^{-x^2}$. Does the limit $\lim_{\mathbf{v} \to \mathbf{0}} g(\mathbf{v})$ exist? If so, what is it. (c) Let $h: \mathbb{R}^2 \to \mathbb{R}$ be given by $h(x,y) = \begin{cases} e^{-x^2}, & y \ge 0\\ 0, & y < 0 \end{cases}$. Does the limit $\lim_{\mathbf{v} \to \mathbf{0}} h(\mathbf{v})$ exist? If so, what is it. (d) Let $\eta: \mathbb{R}^2 \to \mathbb{R}$ be given by $\eta(\mathbf{v}) = e^{-||\mathbf{v}||^2}$. Does the limit $\lim_{\mathbf{v} \to \mathbf{0}} \eta(\mathbf{v})$ exist? If so, what is it.

          (a) Let $f: \mathbb{R} \to \mathbb{R}$ be given by $f(x) = e^{-x^2}$. Does the limit $\lim_{x \to 0} f(x)$ exist? If so, what is it.
(b) Let $g: \mathbb{R}^2 \to \mathbb{R}$ be given by $g(x,y) = e^{-x^2}$. Does the limit $\lim_{\mathbf{v} \to \mathbf{0}} g(\mathbf{v})$ exist? If so, what is it.
(c) Let $h: \mathbb{R}^2 \to \mathbb{R}$ be given by
$h(x,y) = \begin{cases} e^{-x^2}, & y \ge 0\\ 0, & y < 0 \end{cases}$.
Does the limit $\lim_{\mathbf{v} \to \mathbf{0}} h(\mathbf{v})$ exist? If so, what is it.
(d) Let $\eta: \mathbb{R}^2 \to \mathbb{R}$ be given by $\eta(\mathbf{v}) = e^{-||\mathbf{v}||^2}$. Does the limit $\lim_{\mathbf{v} \to \mathbf{0}} \eta(\mathbf{v})$ exist? If so, what is it.

(a) Let f: ℝ→ℝ be given by f(x) = e^-x^2. Does the limit limx → 0 f(x) exist? If so, what is it.
(b) Let g: ℝ^2 →ℝ be given by g(x,y) = e^-x^2. Does the limit lim𝐯→0 g(𝐯) exist? If so, what is it.
(c) Let h: ℝ^2 →ℝ be given by
h(x,y) = e^-x^2, y ≥ 0
0, y < 0.
Does the limit lim𝐯→0 h(𝐯) exist? If so, what is it.
(d) Let η: ℝ^2 →ℝ be given by η(𝐯) = e^-||𝐯||^2. Does the limit lim𝐯→0η(𝐯) exist? If so, what is it.

Added by William B.

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