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Sagot :
Let's evaluate the limit:
[tex]\[ \lim_{x \to 1} \frac{x^{1/4} - 1}{x^{1/3} - 1} \][/tex]
### Step-by-Step Solution:
1. Check the indeterminate form:
First, we substitute [tex]\( x = 1 \)[/tex] directly into the expression to see what form it takes:
[tex]\[ \frac{1^{1/4} - 1}{1^{1/3} - 1} = \frac{1 - 1}{1 - 1} = \frac{0}{0} \][/tex]
The expression is of the indeterminate form [tex]\(\frac{0}{0}\)[/tex], so we need to apply L'Hôpital's Rule.
2. Apply L'Hôpital's Rule:
L'Hôpital's Rule states that if [tex]\(\lim_{x \to c} \frac{f(x)}{g(x)}\)[/tex] gives an indeterminate form [tex]\(\frac{0}{0}\)[/tex] or [tex]\(\frac{\infty}{\infty}\)[/tex], we can evaluate the limit of the derivatives:
[tex]\[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \][/tex]
In this case, [tex]\( f(x) = x^{1/4} - 1 \)[/tex] and [tex]\( g(x) = x^{1/3} - 1 \)[/tex]. We need to find the derivatives of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ f'(x) = \frac{d}{dx} \left( x^{1/4} - 1 \right) = \frac{1}{4} x^{-3/4} \][/tex]
[tex]\[ g'(x) = \frac{d}{dx} \left( x^{1/3} - 1 \right) = \frac{1}{3} x^{-2/3} \][/tex]
3. Evaluate the limit of the derivatives:
Now we have:
[tex]\[ \lim_{x \to 1} \frac{f'(x)}{g'(x)} = \lim_{x \to 1} \frac{\frac{1}{4} x^{-3/4}}{\frac{1}{3} x^{-2/3}} \][/tex]
Simplify the fraction:
[tex]\[ \frac{\frac{1}{4} x^{-3/4}}{\frac{1}{3} x^{-2/3}} = \frac{\frac{1}{4}}{\frac{1}{3}} \cdot \frac{x^{-3/4}}{x^{-2/3}} = \frac{1}{4} \cdot \frac{3}{1} \cdot x^{2/3} \cdot x^{3/4} \][/tex]
Combining the exponents:
[tex]\[ = \frac{3}{4} \cdot x^{\left(-\frac{3}{4} + \frac{2}{3}\right)} = \frac{3}{4} \cdot x^{-\frac{9}{12} + \frac{8}{12}} = \frac{3}{4} \cdot x^{-1/12} \][/tex]
4. Simplify at [tex]\( x \to 1 \)[/tex]:
When [tex]\( x \)[/tex] approaches 1, the term [tex]\( x^{-1/12} \)[/tex] approaches [tex]\( 1 \)[/tex].
So:
[tex]\[ \lim_{x \to 1} \frac{3}{4} \cdot x^{-1/12} = \frac{3}{4} \cdot 1 = \frac{3}{4} \][/tex]
Therefore, the limit is:
[tex]\[ \lim_{x \to 1} \frac{x^{1/4} - 1}{x^{1/3} - 1} = \frac{3}{4} \][/tex]
[tex]\[ \lim_{x \to 1} \frac{x^{1/4} - 1}{x^{1/3} - 1} \][/tex]
### Step-by-Step Solution:
1. Check the indeterminate form:
First, we substitute [tex]\( x = 1 \)[/tex] directly into the expression to see what form it takes:
[tex]\[ \frac{1^{1/4} - 1}{1^{1/3} - 1} = \frac{1 - 1}{1 - 1} = \frac{0}{0} \][/tex]
The expression is of the indeterminate form [tex]\(\frac{0}{0}\)[/tex], so we need to apply L'Hôpital's Rule.
2. Apply L'Hôpital's Rule:
L'Hôpital's Rule states that if [tex]\(\lim_{x \to c} \frac{f(x)}{g(x)}\)[/tex] gives an indeterminate form [tex]\(\frac{0}{0}\)[/tex] or [tex]\(\frac{\infty}{\infty}\)[/tex], we can evaluate the limit of the derivatives:
[tex]\[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \][/tex]
In this case, [tex]\( f(x) = x^{1/4} - 1 \)[/tex] and [tex]\( g(x) = x^{1/3} - 1 \)[/tex]. We need to find the derivatives of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ f'(x) = \frac{d}{dx} \left( x^{1/4} - 1 \right) = \frac{1}{4} x^{-3/4} \][/tex]
[tex]\[ g'(x) = \frac{d}{dx} \left( x^{1/3} - 1 \right) = \frac{1}{3} x^{-2/3} \][/tex]
3. Evaluate the limit of the derivatives:
Now we have:
[tex]\[ \lim_{x \to 1} \frac{f'(x)}{g'(x)} = \lim_{x \to 1} \frac{\frac{1}{4} x^{-3/4}}{\frac{1}{3} x^{-2/3}} \][/tex]
Simplify the fraction:
[tex]\[ \frac{\frac{1}{4} x^{-3/4}}{\frac{1}{3} x^{-2/3}} = \frac{\frac{1}{4}}{\frac{1}{3}} \cdot \frac{x^{-3/4}}{x^{-2/3}} = \frac{1}{4} \cdot \frac{3}{1} \cdot x^{2/3} \cdot x^{3/4} \][/tex]
Combining the exponents:
[tex]\[ = \frac{3}{4} \cdot x^{\left(-\frac{3}{4} + \frac{2}{3}\right)} = \frac{3}{4} \cdot x^{-\frac{9}{12} + \frac{8}{12}} = \frac{3}{4} \cdot x^{-1/12} \][/tex]
4. Simplify at [tex]\( x \to 1 \)[/tex]:
When [tex]\( x \)[/tex] approaches 1, the term [tex]\( x^{-1/12} \)[/tex] approaches [tex]\( 1 \)[/tex].
So:
[tex]\[ \lim_{x \to 1} \frac{3}{4} \cdot x^{-1/12} = \frac{3}{4} \cdot 1 = \frac{3}{4} \][/tex]
Therefore, the limit is:
[tex]\[ \lim_{x \to 1} \frac{x^{1/4} - 1}{x^{1/3} - 1} = \frac{3}{4} \][/tex]
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