Evaluate the following limit, if exist: lim x → 5 (e^x - e⁵)/(x - 5)

Question 9 of 14intermediate🔧 ApplyNumerical2 marks

Evaluate the following limit, if exist: $\lim_{x \to 5} \frac{e^x - e^5}{x - 5}$

Correct Answer

$e^{5}$

Exercise: Exercise 13.2 | Q: 3 | (Page Page 6)

For More Understanding

Explanation

This question from Exercise 13.2 tests the application of the standard limit formula lim(x→0) ( $e^{x}$ - 1)/x = 1. The solution requires substitution to transform the expression into this standard form. The textbook provides the answer as $e^{5}$ , which is derived by factoring out $e^{5}$ and applying the standard exponential limit.

Solution Steps

Step 1: Let x - 5 = h, so x = 5 + h. When x → 5, h → 0.
Step 2: Substitute in the limit: lim(x→5) ( $e^{x}$ - $e^{5}$ )/(x - 5) = lim(h→0) (e^(5+h) - $e^{5}$ )/h
Step 3: Simplify: = lim(h→0) ( $e^{5}$ · $e^{h}$ - $e^{5}$ )/h = lim(h→0) $e^{5}$ ( $e^{h}$ - 1)/h
Step 4: Factor out $e^{5}$ : = $e^{5}$ · lim(h→0) ( $e^{h}$ - 1)/h
Step 5: Apply standard limit lim(x→0) ( $e^{x}$ - 1)/x = 1: = $e^{5}$ · 1 = $e^{5}$

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