Cathy's hesitation in the beginning signifies quite eloquently the paralysing effect that the request for formalism has on novices at this stage. George's reaction is to fully reproduce in (1) and (2) the definition given in the lectures giving however the impression that he does so in a mechanistic, uncritical way: he completely ignores the conditions under which |f(x)-l|<e holds in (1). Then, as a result of the tutor's objection, he restores the conditions impeccably in (2). That the formalistic integrity of G1 and G2 do not reflect the depth of his understanding, is illustrated in G1 which, on one hand, contains the main implication of the definition (x®aÞ f(x)®l) and, on the other hand, shows a very problematic perception of the nature of e. As it will be repeatedly illustrated in this Chapter translating the ® of the above implication into the quantification of d and e is probably the most cognitively problematic aspect of the definition of limit observed in these students.

Moreover C1-C3 are indications of Cathy's semantic misinterpretation of the notation used by the lecturer. In order to denote the dependence of d on a and e, the lecturer used d(a,e) which Cathy mistook, first, for d being a function of a and e (C1) and, then, for an interval (a,e) of which d is an element. In sum, so far Cathy appears to be completely at sea with the concept. The subsequent shift of Cathy's perception of the form and content of the definition of limit seems remarkable and is presented in the following along with further elaboration on the concept. The conversation starts immediately after the tutor has explained the definition of limit (first verbally, in colloquial terms such as 'approaches' and 'close', then formally in (3), introducing d and e, and graphically). While listening to him, Cathy sounds preoccupied with the definition of d (C4).

In C4 Cathy appears as if she is trying to understand the definition of limit from its negation without realising that her words constitute parts of the negation. C5 is a verbalisation of the definition that seems to reflect a growing image of the limiting process as a machine (input: e, output: d ). George also accentuates this with 'find a d' in G2 as opposed to 'there would be a d' in C5. C6 is more illuminating about Cathy's intentions: she seems to be struggling with the implication

if the limit exists then "e>0 $d>0...

and wants to find out if it is likely at all for the limit to exist and at the same time to be unable to find a d. The tutor (T5) still thinks she is trying to construct the negation but she seems surprised at this interpretation (C7). Her surprise in C7 might also be attributed to the common belief that definitions are undeniable (by 'denying' the tutor possibly refers to negating the categorical proposition contained in the definition). Then in C8 it seems that, for the time, Cathy has resolved her perplexity at the definition of limit. Her 'close enough' verbalisation in C8 is an articulate one and is enthusiastically received by the tutor. However Cathy's enlightenment proves very short-lived.

Led by the tutor through applying the definition of limit on f(x)=x, Cathy appears capable of finding a d on her own (I note that whereas e is given by the tutor as equal to 10^-472 she replies 'take d=e' which shows that she has generalised the process of finding d and doesn't need the specificity of 10^-472). Moreover both students appear to be beginning to comprehend the mechanism of the definition as seen in their observations on d and its dependence on e and the function.

Subsequently the students remain silent even though at first Cathy simplifies the definition of CD4.1i and the tutor gives them a graph of the function in CD4.1i. I note here, even though the only evidence is the students' silence, that this shift from participation to passivity might be attributed to their unfamiliarity with piece-wise functions and to the tutor's use of the theorem that lim_x®af(x) exists if and only if lim_x®a+f(x)= lim_x®a-f(x). Despite his tacit use of the theorem, the tutor then changes again to using the definition for a 'proper proof': to prove that the limit does not exist one has to show that, "LÎÂ, L¹lim_x®a-f(x), i.e.,

if LÎÂ: $e>0 such that: "d>0, 0<|x-a|<dÞ |f(x)-L|>e

Cathy's 'in terms of epsilon' shows puzzlement and her 'does L have to be -1 and 1?' sounds reminiscent of her persistent image of all functions being continuous. She also sounds as if she has missed the argument that the definition must be negated "LÎÂ and thus she is struggling with taking specific values of L. George (G3) in general sounds more at ease with the argument: he can see that for L>1, L then obviously cannot be the limit because there are no values of the function at all around it and he also contributes with a better offer for e.

C10 and C11 are peculiar statements in which the student appears to have despatched her own statement 'as x gets close enough to 0, f(x) gets close enough to 0' from the definition of limit. With the tutor's reassurance she restores the connection but, even though she declares her intention to find a d, it is George who suggests d=e.

In CD4.1v the tutor is unpleasantly surprised by the students' inability to picture xsin1/x even though George's observation that -1£sinx£1 is a good start. Only when given the full picture, Cathy strikes a good guess for the limit of xsin1/x and a fair, if not successful, attempt at choosing d (even though her d is not the optimal choice, the mechanisms of her choice appear to have improved remarkably: she seems to be solving xsin1/x in terms of x and then suggesting d=x).

A Note on the Tutor's Semantic Interventions in the Students' Writing. The tutor's two interventions in the students' writing ((1), (2), (3), (4) in Extract 7.1) highlight the need to foster in the novices a flair for a creative use of notational as well as lexical language: he replaces the ambiguous d(a, e) with 0<|x-a|<d in George's definition of limit and he adds the words 'suppose' and 'then' in Cathy's proof for CD4.1ii. Especially in the latter case his interventions accentuate the implication contained in the definition of limit and can be seen as part of the tutor's encouragement of his students to use language in order to punctuate and clarify mathematical statements.

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