Explain the relationship between the limits to presidential power and the importance of the president's informal powers.

Presidential Powers
Before we discuss limits on president power and how that connects to informal powers, it is helpful to understand what the president can do, according to the Constitution:
The President can execute (carry out) the laws made by Congress
The President is commander in chief of the Army and Navy of the United States
The President has the power to grant pardons for offenses, except in cases of impeachment
The President can make treaties with 2/3rds consent of the Senate
The President can nominate ambassadors, Supreme Court justices, cabinet officers and other executive officers with consent of the Senate
Limits to Presidential Power
The President's powers are limited by the system of checks and balances present in our government.  This system is also written into the Constitution.  The Legislative branch can overrule a Presidential veto with a 2/3rds vote, stop budgeting, and impeach the President.  The Judicial branch cannot be tampered with; once appointed, judges sit in their position for life.  The Courts can also use judicial review, which examines actions by the Legislature or Executive branch for constitutionality.  If the Courts find an action unconstitutional, they can reverse it.
Connections to Informal Powers
The President's informal powers are:
Setting the agenda for Congress
Issuing executive orders
Sending out troops without a declaration of war
Conducting initiatives and actions concerning foreign policy
How are these informal powers connected to the limits and restrictions on formal Presidential power?  
Since the Legislative branch has the power to override and veto Presidential action, the President can use his informal power of setting the agenda wisely- if the agenda does not contain any hot button issues, or if the President waits until he has a Congress that shares his partisan views, he can assume that his actions will be accepted or passed.  It requires a 2/3rds vote in both the House and Senate to override Presidential action.  
The President has the power to issue executive orders, a rule or law issued DIRECTLY from the President without going through Congress.  This bypasses the lengthy process of turning a bill into law.  However, it is important to note that is the executive order is unconstitutional, then the Supreme Court can overrule it.  
The President has the ability to send troops abroad without waiting for the Senate to formally declare war.  This has been used several times in the recent past with the understanding that immediate action needed to take place before the Senate could vote on declaring war, like when George W. Bush sent troops to Afghanistan in the wake of the 9/11 terror attacks.
The President's information powers are a way to bypass some of the checks on Executive power, but they do not give the President free reign as head of the country.
 
http://law2.umkc.edu/faculty/projects/ftrials/conlaw/prespowers.html