Calculus Examples

$lim_{x \to - 2 R} \frac{2 x + 3}{x + 2}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $- 2 R$ .

$\frac{lim_{x \to - 2 R} 2 x + 3}{lim_{x \to - 2 R} x + 2}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2 R$ .

$\frac{lim_{x \to - 2 R} 2 x + lim_{x \to - 2 R} 3}{lim_{x \to - 2 R} x + 2}$

Step 3

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{2 lim_{x \to - 2 R} x + lim_{x \to - 2 R} 3}{lim_{x \to - 2 R} x + 2}$

Step 4

Evaluate the limit of $3$ which is constant as $x$ approaches $- 2 R$ .

$\frac{2 lim_{x \to - 2 R} x + 3}{lim_{x \to - 2 R} x + 2}$

Step 5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2 R$ .

$\frac{2 lim_{x \to - 2 R} x + 3}{lim_{x \to - 2 R} x + lim_{x \to - 2 R} 2}$

Step 6

Evaluate the limit of $2$ which is constant as $x$ approaches $- 2 R$ .

$\frac{2 lim_{x \to - 2 R} x + 3}{lim_{x \to - 2 R} x + 2}$

Step 7

Evaluate the limits by plugging in $- 2 R$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $- 2 R$ for $x$ .

$\frac{2 (- 2 R) + 3}{lim_{x \to - 2 R} x + 2}$

Step 7.2

Evaluate the limit of $x$ by plugging in $- 2 R$ for $x$ .

$\frac{2 (- 2 R) + 3}{- 2 R + 2}$

Step 8

Simplify the answer.

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Step 8.1

Multiply $- 2$ by $2$ .

$\frac{- 4 R + 3}{- 2 R + 2}$

Step 8.2

Factor $2$ out of $- 2 R + 2$ .

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Step 8.2.1

Factor $2$ out of $- 2 R$ .

$\frac{- 4 R + 3}{2 (- R) + 2}$

Step 8.2.2

Factor $2$ out of $2$ .

$\frac{- 4 R + 3}{2 (- R) + 2 (1)}$

Step 8.2.3

Factor $2$ out of $2 (- R) + 2 (1)$ .

$\frac{- 4 R + 3}{2 (- R + 1)}$

Step 8.3

Factor $- 1$ out of $- 4 R$ .

$\frac{- (4 R) + 3}{2 (- R + 1)}$

Step 8.4

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{- (4 R) - 1 (- 3)}{2 (- R + 1)}$

Step 8.5

Factor $- 1$ out of $- (4 R) - 1 (- 3)$ .

$\frac{- (4 R - 3)}{2 (- R + 1)}$

Step 8.6

Rewrite $- (4 R - 3)$ as $- 1 (4 R - 3)$ .

$\frac{- 1 (4 R - 3)}{2 (- R + 1)}$

Step 8.7

Factor $- 1$ out of $- R$ .

$\frac{- 1 (4 R - 3)}{2 (- (R) + 1)}$

Step 8.8

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (4 R - 3)}{2 (- (R) - 1 (- 1))}$

Step 8.9

Factor $- 1$ out of $- (R) - 1 (- 1)$ .

$\frac{- 1 (4 R - 3)}{2 (- (R - 1))}$

Step 8.10

Rewrite $- (R - 1)$ as $- 1 (R - 1)$ .

$\frac{- 1 (4 R - 3)}{2 (- 1 (R - 1))}$

Step 8.11

Cancel the common factor.

$\frac{- 1 (4 R - 3)}{2 (- 1 (R - 1))}$

Step 8.12

Rewrite the expression.

$\frac{4 R - 3}{2 (R - 1)}$