Thread: limits question

Let f (x) be a function defined for all x, with −5 ≤ f (x) ≤ 10. Also, lim f(x) as x ->0 does not exist but f (0) = 3.

(a) Let g (x) = xf(x). Show g is continuous at x = 0.
i know that for the above i have to use the 3 conditions
f(c) exists
lim f(x) as x->0- = lim f(x) as x -> 0+
lim f(x) as x -> 0 = f(c)

(b) Does the graph of g have a tangent line at (0, 0)? Explain.

for this one i'm not sure because g(x)= xf(x)
so g'(x) = f(x) + xf'(x)
g'(x) = 3 + 0 * f'(x) [ i'm not sure about the value of f'(x)
can anyone help me with the value of f'(x) and whether g(0) has a tangent line

Originally Posted by ubhutto

(a) Let g (x) = xf(x). Show g is continuous at x = 0.
i know that for the above i have to use the 3 conditions
f(c) exists
lim f(x) as x->0- = lim f(x) as x -> 0+
lim f(x) as x -> 0 = f(c)

I assume the three conditions should be formulated for g and not f. Obviously, g(0) = 0. For the other two conditions, it is easier to apply the ε-δ definition of continuity. Suppose you are given an ε > 0. Which δ > 0 can you choose so that |x - 0| < δ would guarantee that |g(x) - g(0)| < ε? Note that |g(x) - g(0)| = |x * f(x)| = |x| * |f(x)| and |f(x)| ≤ 10.

Originally Posted by ubhutto

(b) Does the graph of g have a tangent line at (0, 0)? Explain.

for this one i'm not sure because g(x)= xf(x)
so g'(x) = f(x) + xf'(x)
g'(x) = 3 + 0 * f'(x) [ i'm not sure about the value of f'(x)
can anyone help me with the value of f'(x) and whether g(0) has a tangent line

You can only apply the product rule when both functions are differentiable at the given point. Here f(x) is not even continuous at 0. Try using the definition of derivative as a limit.