ことができる

1. 1Ability: the subject has the skill or capacity to do the action.Ex. 1
2. 2Possibility: the situation, place or rules make the action possible or permitted.Ex. 2

A neutral way to say 'can'. It is somewhat more formal and explicit than the potential verb form (e.g. 弾ける, 食べられる), so it is common in writing, explanations, notices and polite speech. It expresses either ability or whether circumstances allow an action. It is longer and stiffer than the potential form. Unlike the られる form of ichidan verbs, it cannot be read as passive or honorific.

• られるN3Both senses of 「られる」 correspond to 「ことができる」, and they are mostly interchangeable, but 「られる」 is more conversational; it attaches only to ichidan verbs and 「来る」, godan verbs use the potential form instead, and 「する」 uses 「できる」; the object is often marked with 「が」, and the same form can also express passive and respect.
• ようになるN4「ことができる」 only states whether something can or cannot be done; 「ようになる」 expresses a change from being unable to being able (usually attaching to the potential form, as in 「乗れるようになった」), and can also express acquiring a new habit. The two are not interchangeable, but can be combined as 「ことができるようになる」.
• 得るN2「得る」 attaches to the masu-stem and is literary and stiff. It is interchangeable with 「ことができる」 when something is conditionally possible; however, for the possibility of a situation occurring (as in 「起こり得る」), 「得る」 is preferred, while personal skills (such as being able to swim) do not take 「得る」.
• 得ないN2「得ない」 attaches to the masu-stem, is literary, and asserts that something is logically or conditionally impossible (as in 「あり得ない」). It is close to 「ことができない」 only in the sense of being unable to do or conceive of something, but is stiffer and more emphatic; in everyday speech, 「〜られない」 is more common.
• かねるN2「かねる」 attaches to the masu-stem and means something is hard to do due to one's position, rules, propriety, or feelings—not a matter of ability; it is formal and euphemistic, often used in refusals. For lack of ability, use 「ことができない」 or the negative potential form, not 「かねる」.

られる is the potential ending for ichidan verbs (godan verbs use forms like 弾ける) and can also be passive or honorific; ようになる describes a change into being able, whereas ことができる simply states ability or possibility. 得る is a formal way to say something can happen, 得ない denies any possibility, and かねる politely says one is unable or unwilling to do something, often as a refusal.