Calculus Examples

$lim_{x \to 1} \frac{1 - \sqrt[y]{x}}{1 - \sqrt[z]{x}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $1$ .

$\frac{lim_{x \to 1} 1 - \sqrt[y]{x}}{lim_{x \to 1} 1 - \sqrt[z]{x}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{lim_{x \to 1} 1 - lim_{x \to 1} \sqrt[y]{x}}{lim_{x \to 1} 1 - \sqrt[z]{x}}$

Step 3

Evaluate the limit of $1$ which is constant as $x$ approaches $1$ .

$\frac{1 - lim_{x \to 1} \sqrt[y]{x}}{lim_{x \to 1} 1 - \sqrt[z]{x}}$

Step 4

Move the limit under the radical sign.

$\frac{1 - \sqrt[y]{lim_{x \to 1} x}}{lim_{x \to 1} 1 - \sqrt[z]{x}}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $1$ .

$\frac{1 - \sqrt[y]{lim_{x \to 1} x}}{lim_{x \to 1} 1 - lim_{x \to 1} \sqrt[z]{x}}$

Step 6

Evaluate the limit of $1$ which is constant as $x$ approaches $1$ .

$\frac{1 - \sqrt[y]{lim_{x \to 1} x}}{1 - lim_{x \to 1} \sqrt[z]{x}}$

Step 7

Move the limit under the radical sign.

$\frac{1 - \sqrt[y]{lim_{x \to 1} x}}{1 - \sqrt[z]{lim_{x \to 1} x}}$

Step 8

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{1 - \sqrt[y]{1}}{1 - \sqrt[z]{lim_{x \to 1} x}}$

Step 8.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{1 - \sqrt[y]{1}}{1 - \sqrt[z]{1}}$