A debt ceiling is a limit on how much a government can legally borrow through the issuance of bonds. This limit is usually set by a government's legislative branch as a way to prevent reckless issuance of government bonds. Once the debt limit is reached, congress would set up a series of sessions whether to increase the limit or not. If a tax cut is imposed to encourage economic growth, the lost revenues from the cut can potentially put a government's fiscal budget on a deficit if expenditures are rising. Once a deficit is underway, the government will have to borrow the additional funds through the issuance of treasuries (treasury bills, notes, bonds). During a deficit, the funds generated through the sale of bonds adds up to the public debt of the government.
Lets say the US government has set a debt ceiling of $14 trillion and its current public debt is $11 trillion. If the total annual expenditure is $3 trillion and only $1.5 trillion is collected from taxes as a result of several tax cuts, the government would have to borrow the extra $1.5 trillion from bonds to keep operating. This scenario is what we call a fiscal deficit. If the government can collect $3.5 trillion from taxes, it is running a surplus since its expenditures is only $3 trillion and still has an extra 500 billion to pay and lower the public debt. Now, if the US ran a deficit for 2 years, and its expenditures remained at $3 trillion, then it would have accumulated an extra $3 trillion from issuing bonds ($1.5 trillion a year X 2 years). From a current public debt of $11 trillion, the $3 trillion incurred from borrowing will now hit the debt ceiling at $14 trillion. Once this is reached, congress would have to decide if they will increase the debt ceiling so they can keep borrowing the needed $1.5 trillion to cover the $3 trillion in expenditures. If the debt ceiling is not raised, the government would not be able to fulfill their obligations worth $1.5 trillion. It would have to default on its debt that would have a long tern effect on the yield on its future bond issues.
Once a government defaults, its credit rating would be lowered and bond prices will go down which in turn raise its yield. Its economic growth would suffer as a result of expensive credit.
To avoid this scenario other than raising the debt ceiling, the government would have to raise taxes to cover the needed funds or lower its expenditure. However, the combination of the three would guarantee continued economic confidence from investors.