How can finite phenomena prove an infinite God?
Can a powerful but finite God be distinguished from an infinitely powerful God? We only have limited resources and powers of observation and generally observe finite phenomena. It appears that infinite phenomena does not exist in reality or we can never hope to verify it as infinite. This is a form of argument from the attributes of God.
"You must prove these pure, unmixed, and uncontrollable attributes [of God] from the present mixed and confused phenomena, and from these alone. A hopeful undertaking! Were the phenomena ever so pure and unmixed, yet being finite, they would be insufficient for that purpose. [...] But there is no view of human life, or of the condition of mankind, from which, without the greatest violence, we can infer the moral attributes [of God], or learn that infinite benevolence, conjoined with infinite power and infinite wisdom"
- "any miracle, feat, or theophany [appearance of God to humans] wrought by an infinite being could—in principle—have also been performed by a finite being. "
- "[The onus of proof is on theists because] the claim they are making is stupendously ambitious: that there exists a being of a type which is massively beyond our personal everyday experience, infinitely powerful, infinitely knowledgeable, perfectly good, an immaterial mind that can act instantaneously and without needing any causal intermediaries to do so on the entire universe. "
The infinity of God is standard dogma in most denominations of Christianity and branches of Islam. There is some debate as to if God has actual or potential infinite attributes. Apologists sometimes argue that actual infinities cannot occur as part of the Kalam argument.  When speaking of God, infinite is said to mean "without limit" or "without restriction". 
- Reliably establishing that God has infinite or unlimited attributes requires observation that it is so.
- For finite creatures such as humans, no such observation is possible.
- Therefore, humans cannot distinguish between infinite and non-finite attributes of God.
- Belief in an infinite God is unwarranted.
At most, given appropriate evidence, we might be able to justify an entity with an attribute (such as knowledge or power) with no known limit. This is not the same as "unlimited".
- If God exists, God has certain attributes, such as being perfect and unalterable.
- Objects with these properties are not observed.
- Objects with these properties probably do not exist.
- God probably does not exist.
Other famous expressions of the argument
"The transcendental idea of a necessary and all-sufficient being is so immeasurably great, so high above all that is empirical, which is always conditioned, that we hope in vain to find materials in the sphere of experience sufficiently ample for our conception, and in vain seek the unconditioned among things that are conditioned, while examples, nay, even guidance is denied us by the laws of empirical synthesis. [...] This proof can at most, therefore, demonstrate the existence of an architect of the world, whose efforts are limited by the capabilities of the material with which he works, but not of a creator of the world"
- "First.-"It cannot show that infinity belongs to God. One of the disadvantages, then, or rather a class of disadvantages," (if these had been on the other side they would have been termed proofs of its fallacy and absurdity,) "attending mere a posteriori reasoning, is, that they can never make it appear that infinity belongs in any way to God.
Second.-"It only entitles us to infer the existence of a being of finite extension. The a posteriori argument can only entitle us to infer the existence of a being of finite extension: for by, what rule known in philosophy, can we deduce, from the existence of an effect finite in extent, the existence of a cause of infinity of extension?"
- "The Sender MUST BE Omniscient. ('omniscient' is an adjective meaning to have complete or unlimited knowledge, awareness, and understanding; to perceive all things). Since the density and complexity of the DNA information is billions of times greater than man's present technology, we must conclude that the sender is supremely intelligent and has all knowledge. Again, this is not just a matter of opinion. Nobody denies that DNA contains solid information on a truly mammoth scale. Such deep and comprehensive information has never been located anywhere else. John 16:30. "
Apologists might refer to scriptures to support their view: 
"Jesus looked at them and said. ‘With man this is impossible, but with God all things are possible.’"
"He creates what He pleases. He is All-Knowing, All-Powerful."
- — Surah 30:54 tr. Abul Ala Maududi
Argument from design
- Main Article: Argument from design
- "The infinite power of God has no clearer proof than that furnished by the study and examination of the phenomena of the created universe and the multiple forms and colorations of nature that can never be fully described. "
This is an argument from ignorance and confuses "no known limit" with "limitless".
From first cause
Assuming God is uncaused and non-contingent, he cannot be finite because finite implies contingency. 
This assumes that the first cause argument is valid, which is a questionable assumption.