Precalculus Examples

$lim_{x \to 1} \frac{x^{2} + 1}{x + 1} + x^{2} + 3$

Step 1

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

$lim_{x \to 1} \frac{x^{2} + 1}{x + 1} + lim_{x \to 1} x^{2} + lim_{x \to 1} 3$

Step 2

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

$\frac{lim_{x \to 1} x^{2} + 1}{lim_{x \to 1} x + 1} + lim_{x \to 1} x^{2} + lim_{x \to 1} 3$

Step 3

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

$\frac{lim_{x \to 1} x^{2} + lim_{x \to 1} 1}{lim_{x \to 1} x + 1} + lim_{x \to 1} x^{2} + lim_{x \to 1} 3$

Step 4

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 1} x)}^{2} + lim_{x \to 1} 1}{lim_{x \to 1} x + 1} + lim_{x \to 1} x^{2} + lim_{x \to 1} 3$

Step 5

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

$\frac{{(lim_{x \to 1} x)}^{2} + 1}{lim_{x \to 1} x + 1} + lim_{x \to 1} x^{2} + lim_{x \to 1} 3$

Step 6

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

$\frac{{(lim_{x \to 1} x)}^{2} + 1}{lim_{x \to 1} x + lim_{x \to 1} 1} + lim_{x \to 1} x^{2} + lim_{x \to 1} 3$

Step 7

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

$\frac{{(lim_{x \to 1} x)}^{2} + 1}{lim_{x \to 1} x + 1} + lim_{x \to 1} x^{2} + lim_{x \to 1} 3$

Step 8

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 1} x)}^{2} + 1}{lim_{x \to 1} x + 1} + {(lim_{x \to 1} x)}^{2} + lim_{x \to 1} 3$

Step 9

Evaluate the limit of $3$ which is constant as $x$ approaches $1$ .

$\frac{{(lim_{x \to 1} x)}^{2} + 1}{lim_{x \to 1} x + 1} + {(lim_{x \to 1} x)}^{2} + 3$

Step 10

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

Tap for more steps...

Step 10.1

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

$\frac{1^{2} + 1}{lim_{x \to 1} x + 1} + {(lim_{x \to 1} x)}^{2} + 3$

Step 10.2

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

$\frac{1^{2} + 1}{1 + 1} + {(lim_{x \to 1} x)}^{2} + 3$

Step 10.3

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

$\frac{1^{2} + 1}{1 + 1} + 1^{2} + 3$

Step 11

Simplify the answer.

Tap for more steps...

Step 11.1

Simplify each term.

Tap for more steps...

Step 11.1.1

Simplify the numerator.

Tap for more steps...

Step 11.1.1.1

One to any power is one.

$\frac{1 + 1}{1 + 1} + 1^{2} + 3$

Step 11.1.1.2

Add $1$ and $1$ .

$\frac{2}{1 + 1} + 1^{2} + 3$

Step 11.1.2

Add $1$ and $1$ .

$\frac{2}{2} + 1^{2} + 3$

Step 11.1.3

Divide $2$ by $2$ .

$1 + 1^{2} + 3$

Step 11.1.4

One to any power is one.

$1 + 1 + 3$

Step 11.2

Add $1$ and $1$ .

$2 + 3$

Step 11.3

Add $2$ and $3$ .

$5$