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Vine's Expository Dictionary of NT Words [1]

1: Ἐξαπορέω (Strong'S #1820 — Verb — exaporeo — ex-ap-or-eh'-om-ahee )

is used in the NT in the Passive Voice, with Middle sense, "to be utterly without a way" (ek, "out of," intensive, a, negative, poros, "a way through;" cp. poreuo, "to go through;" (Eng., "ferry" is connected); "to be quite at a loss, without resource, in despair." It is used in  2—Corinthians 1:8 , with reference to life; in  2—Corinthians 4:8 , in the sentence "perplexed, yet not unto (AV, "in') despair," the word "perplexed" translates the verb aporeo, and the phrase "unto despair" translates the intensive form exaporeo, a play on the words. In the Sept.,  Psalm 88:15 , where the translation is "having been lifted up, I was brought low and into despair."

2: Ἀπελπίζω (Strong'S #560 — Verb — apelpizo — ap-el-pid'-zo )

lit., "to hope away" (apo, "away from," elpizo, "to hope"), i.e., "to give up in despair, to despair," is used in  Luke 6:35 , RV, "nothing despairing," i.e., without anxiety as to the result, or not "despairing" of the recompense from God; this is probably the true meaning; AV, "hoping for nothing again." The marg., "of no man," is to be rejected.

King James Dictionary [2]

Despair n.

1. Hopelessness a hopeless state a destitution of hope or expectation.

We are perplexed, but not in despair.  2 Corinthians 4 .

All safety in despair of safety placed.

2. That which causes despair that of which there is no hope.

The mere despair of surgery, he cures.

3. Loss of hope in the mercy of God.

Despair, L To hope. To be without hope to hive up all hope or expectation followed by of.

We despaired even of life.  2 Corinthians 1 .

Never despair of Gods blessings here, or of his reward hereafter.

Charles Buck Theological Dictionary [3]

The loss of hope; that state of mind in which a person loses his confidence in the divine mercy. Some of the best antidotes against despair, says one, may be taken from the consideration,

1. Of the nature of God, his goodness, mercy, &c.

2. The testimony of God: he hath said, he desireth not the death of the sinner.

3. From the works of God: he hath given his Son to die.

4. From his promises,  Hebrews 13:5 .

5. From his command: he hath commanded us to confide in his mercy.

6. From his expostulations, & 100: Baxter on Religious Melancholy; Claude's Essays, p. 388, Robinson's edit.; Gisborne's Sermon on Religious Despondency.

Webster's Dictionary [4]

(1): ( n.) That which is despaired of.

(2): ( n.) Loss of hope; utter hopelessness; complete despondency.

(3): ( v. i.) To be hopeless; to have no hope; to give up all hope or expectation; - often with of.

(4): ( v. t.) To give up as beyond hope or expectation; to despair of.

(5): ( v. t.) To cause to despair.

International Standard Bible Encyclopedia [5]

dē̇ spâr  2 Corinthians 4:8 ex Desperate Job 6:26 Isaiah 17:11 Job 34:6Desperately  Jeremiah 17:9