Evaluate the given limit- displaystyle lim-xrightarrow 0-frac-left - x-1 right -5-1-x-

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Toppr Admin · Accepted Answer

Limx-x2192-0-x-1-5-x2212-1xPut x-1-y So- y-x2192-1 as-xA0-x-x2192-0-limy-x2192-1y5-x2212-15y-x2212-1-5-x22C5-15-x2212-1 -xA0- -xA0- -xA0- -xA0- -xA0-xA0-limx-x2192-axn-x2212-anx-x2212-a-nan-x2212-1-5-x2234-limx-x2192-0-x-5-5-x2212-1x-5

Evaluate the given limit: $lim x \to 0 \frac{{(x + 1)}^{5} - 1}{x}$

$lim x \to 0 \frac{{(x + 1)}^{5} - 1}{x}$
Put $x + 1 = y$
So, $y \to 1$ as $x \to 0$
$= lim y \to 1 \frac{y^{5} - 1^{5}}{y - 1}$
$= 5 \cdot 1^{5 - 1}$ $[lim x \to a \frac{x^{n} - a^{n}}{x - a} = n a^{n - 1}]$
$= 5$
$∴ lim x \to 0 \frac{{(x + 5)}^{5} - 1}{x} = 5$

Evaluate the given limit: limx→0(x+1)5−1x

limx→0(x+1)5−1xPut x+1=y So, y→1 as x→0=limy→1y5−15y−1=5⋅15−1 [limx→axn−anx−a=nan−1]=5∴limx→0(x+5)5−1x=5

Evaluate the given limit: $lim x \to 0 \frac{{(x + 1)}^{5} - 1}{x}$

$lim x \to 0 \frac{{(x + 1)}^{5} - 1}{x}$
Put $x + 1 = y$
So, $y \to 1$ as $x \to 0$
$= lim y \to 1 \frac{y^{5} - 1^{5}}{y - 1}$
$= 5 \cdot 1^{5 - 1}$ $[lim x \to a \frac{x^{n} - a^{n}}{x - a} = n a^{n - 1}]$
$= 5$
$∴ lim x \to 0 \frac{{(x + 5)}^{5} - 1}{x} = 5$